# Perturbed Gaussian generating functions for universal knot invariants

Dror Bar-Natan and Roland van der Veen

###### Abstract

We introduce a new approach to universal quantum knot invariants that
emphasizes generating functions instead of generators and relations. All the
relevant generating functions are shown to be perturbed Gaussians of the form
$Pe^{G}$, where $G$ is quadratic and $P$ is a suitably restricted
”perturbation”. After developing a calculus for such Gaussians in general we
focus on the rank one invariant $\mathbf{Z}_{\mathbb{D}}$ in detail. We
discuss how it dominates the $\mathfrak{sl}_{2}$-colored Jones polynomials and
relates to knot genus and Whitehead doubling. In addition to being a strong
knot invariant that behaves well under natural operations on tangles
$\mathbf{Z}_{\mathbb{D}}$ is also computable in polynomial time in the
crossing number of the knot. We provide a full implementation of the invariant
and provide a table in an appendix.

###### Contents

  1. 1 Introduction
  2. 2 Generating functions for maps between polynomial rings
  3. 3 Turning algebra into linear algebra
    1. 3.1 Heisenberg algebra case
  4. 4 From algebras to tangle invariants 1
    1. 4.1 Tangle diagrams 1
    2. 4.2 Universal tangle invariants 1
    3. 4.3 Computer practicum 1
  5. 5 Main example
    1. 5.1 Two-dimensional algebra and two-step Gaussians
    2. 5.2 Generalities on quasi-triangular Hopf algebras
    3. 5.3 Generalities on the Drinfeld double
    4. 5.4 The construction of the ribbon Hopf algebra $\mathbb{D}$
  6. 6 Gaussian generating functions for $\mathbb{D}$
  7. 7 From algebra to tangle invariants 2
    1. 7.1 Rotational tangle diagrams
    2. 7.2 Universal invariant of rotational tangles
  8. 8 Properties of the knot invariant $\mathbf{Z}_{\mathbb{D}}$
    1. 8.1 The center of $\mathbb{D}$
    2. 8.2 Main results on $Z_{\mathbb{D}}$
    3. 8.3 Computational complexity of the knot invariant $Z_{\mathbb{D}}$
    4. 8.4 Computations using a Seifert surface
    5. 8.5 A closer look at the first order of $Z_{\mathbb{D}}$
    6. 8.6 Computer practicum 2
  9. A Proof of $u=AB\ S(u)$
  10. B Computer implementation
  11. C Glossary of notation
  12. D Table of knots

##  1 Introduction

The key idea of this paper is to replace algebra generators by generating
functions. This approach is especially effective in dealing with quantum
groups and the universal knot invariants that come with them. With some simple
modifications the generating functions for all important operations and
elements take the form of perturbed Gaussians $Pe^{G}$. We will see that
manipulating Gaussians is conceptually and practically simpler than working in
terms of generators.

To illustrate what we mean by replacing generators by generating functions
consider the universal enveloping algebra $H$ of the Heisenberg algebra. It is
an algebra generated by elements $\mathbf{p},\mathbf{x}$ subject to the
relation $\mathbf{p}\mathbf{x}-\mathbf{x}\mathbf{p}=1$. Instead of computing
in this algebra directly we precompute all the multiplications of all the
monomials
$\mathbf{p}^{k_{1}}\mathbf{x}^{n_{1}}\cdot\mathbf{p}^{k_{2}}\mathbf{x}^{n_{2}}$
once and for all and record the results in the following generating function:

$m=\sum_{k_{1},k_{2},n_{1},n_{2}=0}^{\infty}\mathbf{p}^{k_{1}}\mathbf{x}^{n_{1}}\mathbf{p}^{k_{2}}\mathbf{x}^{n_{2}}\frac{\pi^{k_{1}}\xi^{n_{1}}\pi^{k_{2}}\xi^{n_{2}}}{k_{1}!k_{2}!n_{1}!n_{2}!}$

As so often, the generating function is less complicated than the sequential
data it encodes. In this case $m$ has the shape of a Gaussian:

$m=e^{(\pi_{1}+\pi_{2})\mathbf{p}}e^{(\xi_{1}+\xi_{2})\mathbf{x}-\xi_{1}\pi_{2}}$

The generating function technique works especially well for computing
universal knot invariants because there one has to multiply many terms of a
similar exponential form
$\mathbf{R}=e^{\mathbf{p}\otimes\mathbf{x}-\mathbf{1}\otimes\mathbf{p}\mathbf{x}}$.
Roughly speaking the rule is to assign a copy of $\mathbf{R}$ to each crossing
in the knot diagram with the first factor on the overpass and the second on
the underpass. The universal knot invariant $\mathbf{Z}$ is then the product
of all terms taken in order of appearance as one walks along the knot. Using a
quantum field theory inspired calculus of perturbed Gaussian generating
functions we are able to compute the invariant $\mathbf{Z}$ without writing
down a single generator.

Our Gaussian generating function techniques should apply to computations in
many algebras including the universal invariants corresponding to any of the
Drinfeld-Jimbo quantum groups $\mathcal{U}_{\hbar}(\mathfrak{g})$ [14]. For
concreteness however we focus on the rank one case in this paper. We show how
to build a version $\mathbb{D}$ of quantum $\mathfrak{sl}_{2}$ and express all
structural operations and elements in terms of perturbed Gaussian generating
functions.

Expanding our invariant $\mathbf{Z}_{\mathbb{D}}$ as a series in an auxiliary
parameter $\epsilon$ we obtain the Alexander polynomial at the $0$-th order
and increasingly strong invariants as one computes higher orders in
$\epsilon$. Unlike usual quantum knot invariants computations of
$\mathbf{Z}_{\mathbb{D}}(K)$ can be done in polynomial time in the complexity
of the knot $K$, see Theorem 50. Furthermore, we relate
$\mathbf{Z}_{\mathbb{D}}$ to the knot genus, the Alexander polynomial and
Whitehead doubling, see Theorems 51, 54 and 55. In Theorem 52 we show
$\mathbf{Z}_{\mathbb{D}}$ determines all the $\mathfrak{sl}_{2}$ colored Jones
polynomials. We also expect $\mathbf{Z}_{\mathbb{D}}$ is equivalent to
Rozansky’s loop expansion of the colored Jones polynomial [17]. The present
work is a continuation of [BV19] in which a simplified version of
$\mathbf{Z}_{\mathbb{D}}$ appears to first order in $\epsilon$.

In summary $\mathbf{Z}_{\mathbb{D}}$ is a strong yet computable knot invariant
whose topological interpretation is tractable due to its good behaviour under
tangle operations, strand doubling and reversal. As such we hope it may play a
role in addressing questions of topological interest.

The plan of the paper is as follows. First we will introduce Gaussian
generating functions in the context of linear maps between polynomial rings.
In this setting we develop the key contraction theorem that allows us to
compose linear maps by contracting their Gaussian generating functions. Next
we illustrate the main techniques in a section on the Heisenberg algebra and
the universal knot invariants related to it. As a preparation for later
sections we introduce a notion of tangles appropriate for our invariants. The
calculation of the Heisenberg tangle invariants is fully implemented in
Mathematica.

Only in Section 5 we set out to construct our main example of a ribbon Hopf
algebra $\mathbb{D}$ using the Drinfeld double construction. We show in detail
how to compute in $\mathbb{D}$ using Gaussian generating functions in the next
section. After setting up the algebra we extend our notion of tangles to
account for rotation numbers and introduce the universal tangle invariant
$\mathbf{Z}_{\mathbb{D}}$. In Section 8 we prove our main theorems using a
combination of the Gaussian calculus and a Hopf algebraic interpretation of
Seifert surfaces. We end the work with a discussion of the computer
demonstration. In the Appendix B we list the full implementation of our
algorithms and our invariant is tabulated for all prime knots up to ten
crossings in Appendix D.

Acknowledgments.  The authors wish to thank Jorge Becerra for careful reading
of the manuscript and Thomas Fiedler for hosting a workshop on this subject.
D. B. was supported by NSERC grant RGPIN-2018-04350.

##  2 Generating functions for maps between polynomial rings

In this section we describe linear maps between polynomial algebras using
generating functions. The polynomials we will work with111For concreteness we
prefer to work over the rationals although much of what say applies to more
general fields of characteristic $0$. are in $\mathbb{Q}[z_{J}]$ where $J$ is
a finite set and the multi-index $z_{J}$ means $(z_{j})_{j\in J}$. Our goal is
to investigate the linear maps between two such vector spaces. Denote the
space of linear maps by $\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])$.

In later sections many such linear maps come from choosing a basis in a non-
commutative algebra and describing multiplication in terms of the ordered
monomials. Some more elementary examples are the following:

  1. 1.

The identity
$\mathrm{id}_{\mathbb{Q}[z_{J}]}\in\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{J}])$

  2. 2.

The shift map $\mathrm{sh}:\mathbb{Q}[z]\to\mathbb{Q}[z]$ defined by
$\mathrm{sh}(p)(z)=p(z+1)$

  3. 3.

Partial differentiation $\frac{\partial}{\partial
z}:\mathbb{Q}[z,w]\to\mathbb{Q}[z,w]$.

  4. 4.

Definite integration $\int:\mathbb{Q}[z]\to\mathbb{Q}[z]$ defined by $\int
p=\int_{0}^{z}p(t)dt$.

The monomials form a basis for $\mathbb{Q}[z_{J}]$ and so any linear map
$\varphi\in\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])$ is determined by
its values on all monomials. For example the partial derivative can be
characterized by just knowing $\frac{\partial}{\partial
z}z^{k}w^{\ell}=kz^{k-1}w^{\ell}$. In general then our linear map $\varphi$ is
determined by a sequence of polynomials
$\varphi(z_{J}^{n})\in\mathbb{Q}[z_{K}]$ for every multi-index
$n\in\mathbb{N}^{J}$.

This is where the generating functions come into play. Whenever there is a
sequence of numbers or other algebraic objects one can try to understand its
collective behaviour by placing them into a generating function. To encode
each of the basis vectors we use an auxiliary variable $\zeta_{j}$
corresponding to each variable $z_{j}$.

######  Definition 1.

(Generating function of a linear map)
Define
$\mathcal{G}:\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])\to\mathbb{Q}[z_{K}][[\zeta_{J}]]$
by

$\mathcal{G}(\varphi)=\sum_{n\in\mathbb{N}^{J}}\frac{\zeta_{J}^{n}}{n!}\varphi(z_{J}^{n})$

Here $n$ is a multi-index $n=(n_{j})_{j\in J}$ and $n!=\prod_{j\in J}n_{j}!$
and $z_{J}^{n}=\prod_{j\in J}z_{j}^{n_{j}}$.

Notice that in some sense we are just inserting an exponential into $\varphi$
in the above definition. More formally, extending $\varphi$ to a map
$\mathbb{Q}[z_{J}][[\zeta_{J}]]\to\mathbb{Q}[z_{K}][[\zeta_{K}]]$ by treating
the $\zeta_{J}$ as scalars, the exponential generating function may also be
expressed as

$\mathcal{G}(\varphi)=\varphi(e^{z_{J}\zeta_{J}})=\varphi(\mathcal{G}(\mathrm{id}_{\mathbb{Q}[z_{J}]}))$
(1)

This explains why we are using exponential generating functions.

In our partial differentiation example we take $\omega$ to be the variable
corresponding to $w$ and $\zeta$ corresponding to $z$ and find:

$\mathcal{G}(\frac{\partial}{\partial
z})=\sum_{k,\ell}\frac{\partial}{\partial
z}(z^{k}w^{\ell})\frac{\zeta^{k}\omega^{\ell}}{k!\ell!}=\sum_{k,\ell}kz^{k-1}w^{\ell}\frac{\zeta^{k}\omega^{\ell}}{k!\ell!}=e^{\omega
w}\sum_{k}z^{k-1}w^{\ell}\frac{\zeta^{k}}{(k-1)!}=\zeta e^{\omega w+\zeta z}$

The generating functions for the other examples can be found similarly using
properties of the exponential function and perhaps Equation (1):

  1. 1.

$\mathcal{G}(\mathrm{id}_{\mathbb{Q}[z_{J}]})=e^{z_{J}\zeta_{J}}$, where
$z_{J}\zeta_{J}=\sum_{j\in J}z_{j}\zeta_{j}$.

  2. 2.

$\mathcal{G}(\mathrm{sh})=e^{(z+1)\zeta}$

  3. 3.

$\mathcal{G}(\frac{\partial}{\partial z})=\zeta e^{\omega w+\zeta z}$.

  4. 4.

$\mathcal{G}(\int)=\frac{e^{\zeta z}-1}{\zeta}$.

The polynomial ring is a Hopf algebra and this provides us with a few more
examples of linear maps. Many of the generating functions we will use later to
describe more general Hopf algebras have features similar to the simple ones
presented here. The Hopf algebra maps and their generating functions are
listed below:

  1. 1.

Multiplication $m:\mathbb{Q}[z_{1},z_{2}]\to\mathbb{Q}[z],\quad
m(z_{1}^{k}z_{2}^{\ell})=z_{1}^{k}z_{2}^{\ell}$,
$\mathcal{G}(m)=e^{(\zeta_{1}+\zeta_{2})z}$.

  2. 2.

Co-Multiplication $\Delta:\mathbb{Q}[z]\to\mathbb{Q}[z_{1},z_{2}]$,
$\Delta(z^{k})=(z_{1}+z_{2})^{k}$,
$\mathcal{G}(\Delta)=e^{\zeta(z_{1}+z_{2})}$

  3. 3.

Antipode $S:\mathbb{Q}[z]\to\mathbb{Q}[z]$, $S(z^{k})=(-1)^{k}z^{k}$,
$\mathcal{G}(S)=e^{-\zeta z}$

  4. 4.

Unit $\eta:\mathbb{Q}\to\mathbb{Q}[z]$, $\eta(1)=1$, $\mathcal{G}(\eta)=1$

  5. 5.

co-unit $\varepsilon:\mathbb{Q}[z]\to\mathbb{Q}$,
$\varepsilon(z^{k})=\delta_{k,0}$, $\mathcal{G}(\varepsilon)=1$

The generating function contains all information about the linear map and so
we can move back and forth between the two descriptions.

######  Lemma 2.

(The generating function determines the linear map)
$\mathcal{G}:\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])\to\mathbb{Q}[z_{K}][[\zeta_{J}]]$
is a bijection. The inverse is given by

$\mathcal{G}^{-1}(f)(p)=p(\partial_{\zeta_{J}})f(\zeta_{J},z_{K})|_{\zeta_{J}=0}=f(\partial_{z_{J}},z_{K})p(z_{J})|_{z_{J}=0}$

where $p\in\mathbb{Q}[z_{J}]$ and $f\in\mathbb{Q}[z_{K}][[\zeta_{J}]]$.

###### Proof.

Notice that $p(z_{J})=p(\partial_{\zeta_{J}})e^{\zeta_{J}\cdot
z_{J}}|_{\zeta_{J}=0}$. Therefore

$(\mathcal{G}^{-1}\mathcal{G}(\varphi))(p)=p(\partial_{\zeta_{J}})\varphi(e^{\zeta_{J}\cdot
z_{J}})|_{\zeta_{J}=0}=\varphi(p(\partial_{\zeta_{J}})e^{\zeta_{J}\cdot
z_{J}}|_{\zeta_{J}=0})=\varphi(p)$

Likewise for $f(\zeta_{J},z_{K})$ we find

$\mathcal{G}(\mathcal{G}^{-1}(f))=(\mathcal{G}^{-1}f)(e^{\zeta_{J}\cdot
z_{J}})=f(\partial_{z_{J}},z_{K})e^{\zeta_{J}\cdot
z_{J}}|_{z_{J}=0}=f(\zeta_{J},z_{K})$

∎

The advantage of the generating function approach is in describing composition
of linear maps. Composition222In what follows we will often write composition
as $f\circ g=g//f$. translates into the following operation on the
corresponding generating functions.

######  Lemma 3.

(Composition of generating functions)
Suppose $J,K,L$ are finite sets and
$\varphi\in\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])$ and
$\psi\in\mathrm{Hom}(\mathbb{Q}[z_{K}],\mathbb{Q}[z_{L}])$. We have

$\mathcal{G}(\varphi//\psi)=\Big{(}\mathcal{G}(\varphi)|_{z_{K}\mapsto\partial_{\zeta_{K}}}\mathcal{G}(\psi)\Big{)}|_{\zeta_{K}=0}$

###### Proof.

Set $\alpha=\mathcal{G}(\varphi)$ and $\beta=\mathcal{G}(\psi)$ then

$\mathcal{G}(\varphi//\psi)=e^{\zeta_{K}\cdot
z_{K}}//\varphi//\psi=\alpha//\psi=\alpha//\mathcal{G}^{-1}(\beta)=\Big{(}\alpha|_{z_{K}\mapsto\partial_{\zeta_{K}}}\beta\Big{)}|_{\zeta_{K}=0}$

using Lemma 2 in the final equality. ∎

In categorical terms we can summarize what we found so far by introducing two
categories $\tilde{\mathcal{P}}$ and $\tilde{\mathcal{C}}$ whose objects are
finite sets and whose morphisms are
$\mathrm{Hom}_{\tilde{\mathcal{P}}}(J,K)=\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{K}])$
and $\mathrm{Hom}_{\tilde{\mathcal{C}}}(J,K)=\mathbb{Q}[z_{K}][[z_{J}]]$.
Composition in $\tilde{\mathcal{C}}$ is defined as in the previous Lemma:

$f//g=\Big{(}f|_{z_{K}\mapsto\partial_{\zeta_{K}}}g\Big{)}|_{\zeta_{K}=0}\qquad
f\in\mathrm{Hom}_{\tilde{\mathcal{C}}}(J,K),\
g\in\mathrm{Hom}_{\tilde{\mathcal{C}}}(K,L)$

Then the functor $\mathcal{G}:\tilde{\mathcal{P}}\to\tilde{\mathcal{C}}$ is an
isomorphism of categories. In fact $\mathcal{G}$ is a monoidal functor if we
take $J\otimes K$ to be the disjoint union and $f\otimes g=fg$ in
$\tilde{\mathcal{C}}$.

The composition formula involves differentiating with respect to many
variables at the same time, as many as the size of $K$. We view each of the
$|K|$ differentiations as a contraction, just like one contracts indices in a
tensor. The benefit is that a single contraction operation (defined precisely
below) is simpler to deal with and serves as the fundamental building block
for our computations.

######  Definition 4.

(Contraction)
We say $f$ is the contraction of $g(r,s)=\sum_{k,\ell}c_{k,\ell}r^{k}s^{\ell}$
along the pair of variables $(r,s)$ if

$f=\sum_{k}c_{k,k}k!=\sum_{k,\ell}c_{k,\ell}\partial_{s}^{k}s^{\ell}|_{s=0}$

Our notation will be $f=\langle g(r,s)\rangle_{r,s}$ or simply $f=\langle
g\rangle_{r}$ when it is clear which variables $s$ correspond to the $r$. We
also allow $r$ and $s$ to be vectors of variables of equal size in which case
$r_{i}$ is understood to be contracted with $s_{i}$, so $\langle
g\rangle_{((r_{1},r_{2},r_{3}),(s_{1},s_{2},s_{3}))}=\langle\langle\langle
g\rangle_{(r_{1},s_{1})}\rangle_{(r_{2},s_{2})}\rangle_{(r_{3},s_{3})}$.

So far the variables we used for contractions were $r=\zeta,s=z$. For example
$f=42$ is the contraction of
$g(r,s)=7\zeta_{1}\zeta_{2}^{3}z_{1}z_{2}^{3}+3\zeta_{1}\zeta_{2}^{2}z_{1}^{3}z_{2}^{2}$
with $r=(\zeta_{1},\zeta_{2}),s=(z_{1},z_{2})$:

$\langle g(r,s)\rangle_{r,s}=\langle
7\zeta_{1}\zeta_{2}^{3}z_{1}z_{2}^{3}+3\zeta_{1}\zeta_{2}^{2}z_{1}^{3}z_{2}^{2}\rangle_{(\zeta_{1},\zeta_{2}),(z_{1},z_{2})}=42$

A more subtle example that is central to our theory is about the Gaussian
exponential $g(r,s)=e^{\hbar rs}\in\mathbb{Q}[r,s][[\hbar]]$. Summing the
geometric series shows $f=\frac{1}{1-\hbar}\in\mathbb{Q}[[\hbar]]$ is the
contraction of $g$ along the pair $(r,s)$:

$\langle
g\rangle_{r,s}=\sum_{n=0}^{\infty}\partial_{s}^{n}s^{n}\frac{\hbar^{n}}{n!}=\sum_{n=0}^{\infty}\hbar^{n}=f$

This example also suggests contractions are not always well defined. For
example if we instead take $g=e^{rs}\in\mathbb{Q}[r][[s]]$ then the same
calculation yields $\langle g(r,s)\rangle_{r,s}=\sum_{n=0}^{\infty}1$.

Coming back to our discussion of composition of linear maps by contraction of
their generating functions we arrive at the following. If
$\varphi\in\mathbb{Q}[z_{K}][[\zeta_{J}]]$ and
$\psi\in\mathbb{Q}[z_{L}][[\zeta_{K}]]$ then

$\varphi//\psi=\langle\varphi\psi\rangle_{\zeta_{K},z_{K}}$

where $\varphi\psi$ means ordinary multipliciation of the power series in
$\mathbb{Q}[z_{L},z_{K}][[\zeta_{K},\zeta_{J}]]$.

Looking at the above examples we notice they all involve exponentials of
quadratic forms, also known as _Gaussians_ , possibly with some kind of
perturbation. Perturbed Gaussians not only appear in all our examples, there
is also a concrete formula for composing and contracting them!

######  Lemma 5.

(Contraction Lemma)
For any $n\in\mathbb{N}$ consider the ring
$R_{n}=\mathbb{Q}[r_{j},g_{j}][[s_{j},\,\ W_{ij},\ f_{j}|1\leq i,j\leq n]]$.
We have the following equality in $R_{n}$:

$\langle
e^{gs+rf+rWs}\rangle_{r,s}=\det(\tilde{W})e^{g\tilde{W}f}\qquad\tilde{W}=(1-W)^{-1}$

here $r=(r_{i}),s=(s_{i}),g=(g_{i}),f=(f_{i})$ are thought of as vectors of
size $n$ and $W=(W_{ij})$ is an $n\times n$ matrix.

###### Proof.

Since we are in a power series ring $\tilde{W}$ is to be taken as an infinite
series $\tilde{W}_{ab}=\sum_{k=0}^{\infty}W^{k}_{ab}$, where
$W^{k}_{ab}=\sum_{i_{1},\dots i_{k-1}}W_{ai_{1}}W_{i_{1}i_{2}}\dots
W_{i_{k-1}b}$.

We start by proving the special case $f=g=0$. Set $Z(\lambda)=\langle
e^{r\lambda Ws}\rangle_{(r,s)}$ and $A(\lambda)=\frac{1}{\det 1-\lambda W}$.
The result $Z(\lambda)=A(\lambda)$ follows once we show both sides are
solutions to the initial value problem
$\partial_{\lambda}F(\lambda)=(\mathrm{tr}W(1-\lambda W)^{-1})F$ and $F(0)=1$.

For $A(\lambda)$ this is verified using $\det e^{X}=e^{\mathrm{tr}X}$ so

$\partial_{\lambda}A(\lambda)=\partial_{\lambda}\frac{1}{\det 1-\lambda
W}=\partial_{\lambda}e^{-\mathrm{tr}\log(1-\lambda W)}=(\mathrm{tr}W(1-\lambda
W)^{-1})\frac{1}{\det 1-\lambda W}$

For $Z(\lambda)$ we use the shorthand
$\langle\varphi\rangle=\langle\varphi\rangle_{(r,s)}$ and claim that for any
matrix $M$ of size $n$ with entries in $R_{n}$ we have

$\langle rMse^{r\lambda Ws}\rangle=\mathrm{tr}(M)Z(\lambda)+\langle\lambda
rMWse^{r\lambda Ws}\rangle$ (2)

Using Formula (2) repeatedly, we find a geometric series:

$\partial_{\lambda}Z(\lambda)=\langle rWse^{r\lambda
Ws}\rangle=(\mathrm{tr}W)Z(\lambda)+\langle r\lambda W^{2}se^{r\lambda
Ws}\rangle=$ $(\mathrm{tr}W)Z(\lambda)+(\mathrm{tr}\lambda
W^{2})Z(\lambda)+\langle r\lambda^{2}W^{3}se^{r\lambda
Ws}\rangle=\dots=(\mathrm{tr}W(1-\lambda W)^{-1})Z(\lambda)$

the error term vanishes because the powers of $W$ converge to $0$ in the
topology of $R^{n}$. Finally, formula (2) is verified explicitly by expanding
both sides as power series in $r,s$ and carrying out the contraction in terms
of monomials. This proves $A(\lambda)=Z(\lambda)$ and hence the special case
$f=g=0$.

For the general case we follow a similar strategy. Introduce

$X(\lambda)=\langle e^{rWs+\lambda(gs+rf)}\rangle,\quad
B(\lambda)=\det(1-W)^{-1}e^{\lambda^{2}g(1-W)^{-1}f}$

and show that $\partial_{\lambda}B(\lambda)=\partial_{\lambda}X(\lambda)$.
This is enough since equality at $\lambda=0$ is precisely the special case
$f=g=0$ above. The key is the following equation for any vectors $F,G$ of
length $n$ in $R_{n}$ (proven in the same way as Equation (2)):

$\langle(Gs+rF)e^{rWs+\lambda(gs+rf)}\rangle=2\lambda
GFX(\lambda)+\langle(GWs+rWF)e^{rWs+\lambda(gs+rf)}\rangle$ (3)

Applying this equation repeatedly and using the geometric series finishes the
proof:

$\partial_{\lambda}X(\lambda)=\langle(gs+rf)e^{rWs+\lambda(gs+rf)}\rangle=2\lambda
gfX(\lambda)+\langle(gWs+rWf)e^{rWs+\lambda(gs+rf)}\rangle=$
$2\lambda(gf+gWf)X(\lambda)+\langle(gW^{2}s+rW^{2}f)e^{rWs+\lambda(gs+rf)}\rangle=\dots=2\lambda
g(1-W)^{-1}fX(\lambda)=\partial_{\lambda}B(\lambda)$

∎

For later use we note that we may allow more general perturbations:

######  Theorem 6.

(Contraction Theorem)
With the same notation as in Lemma 5 and $P\in R_{n}$ that only depends on
$r,s$. We have the following equality:

$\langle
P(r,s)e^{gs+rf+rWs}\rangle_{r,s}=\det(\tilde{W})e^{g\tilde{W}f}\langle
P(r+g\tilde{W},\tilde{W}(s+f))\rangle_{r,s}\qquad\tilde{W}=(1-W)^{-1}$

###### Proof.

To derive the theorem from Lemma 5 above we introduce auxiliary variables
$m,\mu$ and write

$\langle P(r,s)e^{gs+rf+rWs}\rangle=P(\partial_{m},\partial_{\mu})\langle
e^{(g+\mu)s+r(f+m)+rWs}\rangle|_{m=\mu=0}$

since these differentiations commute with contraction. Replacing $f$ by $f+m$
and $g$ by $f+\mu$ Lemma 5 says

$\langle
P(r,s)e^{gs+rf+rWs}\rangle=\det(\tilde{W})P(\partial_{m},\partial_{\mu})e^{(g+\mu)\tilde{W}(f+m)}|_{m=\mu=0}=$
$\det(\tilde{W})P(\partial_{m},\tilde{W}(f+m))e^{g\tilde{W}(f+m)}|_{m=0}=\det(\tilde{W})e^{g\tilde{W}f}\langle
P(r+g\tilde{W},\tilde{W}(f+s))\rangle$

∎

Often one can just contract one variable at the time and for that the
following simplified version of Theorem 6 is useful:

$\langle P(r,s)e^{c+rf+gs+Wrs}\rangle_{r,s}=\langle
P\big{(}r+\frac{g}{1-W},\frac{(s+f)}{1-W}\big{)}\rangle_{r,s}\frac{e^{c+\frac{gf}{1-W}}}{1-W}$
(4)

However if one needs to bound the degree of the denominators appearing after
contraction it is better to use the full power of the theorem instead of
repeatedly using the one variable case.

Finally we remark that contraction may also be phrased more symmetrically as

$\langle g(r,s)\rangle_{r,s}=e^{\partial_{r}\partial_{s}}g(r,s)|_{r=s=0}$

This helps seeing the close relationship with formal Gaussian integration,
especially as it is used in physics [15, 1]. In perturbation theory, without
paying attention to convergence issues, there is another form for the
contraction:

$\langle g(r,s)\rangle_{r,s}\propto\int e^{-rs}g(r,s)drds$

Many of the computations in this paper may be recast in terms of the above
formula and Gaussian integration, yet we prefer to use this fact only for
inspiration. There is simply nothing to gain: everything one can do with
integration we can also do directly with (4). The perspective of integration
does sometimes make it clearer perturbed Gaussians are closed under
contraction.

##  3 Turning algebra into linear algebra

In this section we lay the foundation for our application of generating
function techniques to computations in algebra. For the sake of argument we
will set up our construction for algebras $A$ over $\mathbb{Q}$ but later the
same ideas will be applied slightly more generally.

In analogy with the common notation for polynomial rings $\mathbb{Q}[z_{J}]$
for a set $J$ we notation we will use something similar for tensor products of
more general algebras. In this notation the factors in the tensor product are
indexed by the elements of $J$. The subscript indicates the tensor factor an
element is in.

######  Definition 7.

(Labelled tensor products)
For any finite set $J$ and any algebra $A$ with unit we set $A^{\otimes J}$ to
be the free algebra on elements $\\{a_{j}|a\in A,j\in J\\}$ quotiented out by
the relations $a_{j}a^{\prime}_{j}=(aa^{\prime})_{j}$ for $a,a^{\prime}\in A$
and $a_{i}a^{\prime}_{j}=a^{\prime}_{i}a_{j}$ for all $i\neq j$. Any factors
$1_{j}$ will be ommitted. We identify $A^{\otimes\\{1,2,\dots n\\}}$ with
$A^{\otimes n}$ sending $a_{i}$ to $1\otimes 1\otimes\dots a\otimes
1\dots\otimes 1$ with $a$ in the $i$-th position.

For example we would write $1\otimes x+x\otimes 1+x\otimes
y=x_{2}+x_{1}+x_{1}y_{2}\in A^{\otimes\\{1,2\\}}$.

We say $A$ is a PBW333The name refers to the Poincare-Birkhoff-Witt theorem in
Lie theory that says that the universal enveloping algebra is PBW in the above
sense. algebra if there exists a vector space isomorphism
$\mathbb{O}:\mathbb{Q}[z]\to A$. The variable $z$ is allowed to be a vector of
variables. For example the associative algebra $A=\mathcal{U}(\mathfrak{h})$
generated $\mathbf{p},\mathbf{x}$, subject to the relation
$[\mathbf{p},\mathbf{x}]=1$ is PBW. We will use the isomorphism
$\mathbb{O}:\mathbb{Q}[p,x]\to A$ that sends the commutative monomials to
lexicographically ordered monomial in the generators $z=(p,x)$. More precisely
we will use the vector space isomorphism
$\mathbb{O}:\mathbb{Q}[p,x]\to\mathcal{U}(\mathfrak{h})$ sending
$p^{k}x^{\ell}$ to $\mathbf{p}^{k}\mathbf{x}^{\ell}$.

In a PBW algebra $A$ we will use the same notation $\mathbb{O}$ also for its
extensions to tensor powers, so $\mathbb{O}:\mathbb{Q}[z_{J}]\to A^{\otimes
J}$. With these definitions in place we can use $\mathbb{O}$ to transfer any
structures on $A$ and its tensor powers to a linear map between polynomial
rings. Recall we introduced a category $\tilde{\mathcal{P}}$ for such linear
maps. If we define a similar category $\tilde{\mathcal{H}}$ for morphisms
between tensor powers of $A$ then $\mathbb{O}$ gives us an equivalence of
categories.

######  Lemma 8.

Define a category $\tilde{\mathcal{H}}$ whose objects are finite sets and
whose morphisms are linear maps $\mathrm{Hom}(A^{\otimes J},A^{\otimes K})$.
The isomorphism $\mathbb{O}$ gives rise to an isomorphism of monoidal
categories $\mathcal{O}:\tilde{\mathcal{H}}\to\tilde{\mathcal{P}}$ defined by
$\mathcal{O}(f)=\mathbb{O}^{-1}//f//\mathbb{O}$.

Combining with the generating function functor $\mathcal{G}$ gives us a way to
describe many structures on $A$ using generating functions:

$\tilde{\mathcal{H}}\xrightarrow{\mathcal{O}}\tilde{\mathcal{P}}\xrightarrow{\mathcal{G}}\tilde{\mathcal{C}}$

where we recall all three categories have finite sets as objects and
$\mathrm{Hom}_{\tilde{\mathcal{C}}}(J,K)=\mathbb{Q}[z_{K}][[\zeta_{J}]]$ and
$\mathrm{Hom}_{\tilde{\mathcal{P}}}(J,K)=\mathrm{Hom}(\mathbb{Q}[z_{J}],\mathbb{Q}[z_{J}])$.

The prime example of the type of map we would like to describe from the point
of view of the polynomial ring is the multiplication itself. In our notation
it makes sense to expand the algebra multiplication map $m:A\otimes A\to A$ to
a family of maps

$\mathbf{m}^{ij}_{k}:A^{\otimes\\{i,j\\}}\to A^{\otimes\\{k\\}}$

where $\mathbf{m}^{ij}_{k}$ multiplies from the $i$-th tensor factor with the
$j$-th tensor factor and places the result in tensor factor indexed $k$.

${A^{\otimes\\{i,j\\}}}$${A^{\otimes\\{k\\}}}$${\mathbb{Q}[z_{i},z_{j}]}$${\mathbb{Q}[z_{k}]}$$\scriptstyle{\mathbf{m}^{ij}_{k}}$$\scriptstyle{m^{ij}_{k}}$$\scriptstyle{\mathbb{O}}$$\scriptstyle{\mathbb{O}}$

As a notational convention we write elements and maps from the original
algebra in boldface, polynomial ring objects in regular script and generating
functions either as $\mathcal{G}(f)$ or as a smaller pre-superscript. For
example $\mathbf{m}^{ij}_{k}$, $m^{ij}_{k}$ and generating function
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ij}_{k}=\mathcal{G}(m^{ij}_{k})$.
In the next subsection we will see some concrete instances of this process.

###  3.1 Heisenberg algebra case

To further illustrate our technique of turning algebra into linear algebra and
then into generating functions we focus on the Heisenberg algebra. Recall this
is $A=\mathcal{U}(\mathfrak{h})$ with generators $\mathbf{p,x}$ and relation
$[\mathbf{p,x}]=1$. We ordered the generators in each monomial alphabetically
to obtain a vector space isomorphism $\mathbb{O}:\mathbb{Q}[p,x]\to A$. In
this example we take $z=(p,x)$.

The way we will apply our generating function techniques to carry out
computations in algebras is by choosing a basis. Throughout this section we
will assume we have an algebra $A$ together with a vector space isomorphism
$\mathbb{O}:\mathbb{Q}[z]\to A$. For example we can consider the associative
algebra $A=\mathcal{\mathfrak{h}}$ generated $\mathbf{p},\mathbf{x}$, subject
to the relation $[\mathbf{p},\mathbf{x}]=1$. Like in all universal enveloping
algebras of Lie algebras the PBW theorem tells us there is a vector space
isomorphism $\mathbb{O}:\mathbb{Q}[p,x]\to A$ sending a monomial to an ordered
one. So in this case $z=(p,x)$ and we choose to use alphabetic ordering of the
monomials. More precisely we will use the vector space isomorphism
$\mathbb{O}:\mathbb{Q}[p,x]\to\mathcal{U}(\mathfrak{h})$ sending
$p^{k}x^{\ell}$ to $\mathbf{p}^{k}\mathbf{x}^{\ell}$.

######  Lemma 9.

(Generating function for multiplication)

$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ij}_{k}=\mathcal{G}(m^{ij}_{k})=e^{(\pi_{i}+\pi_{j})p_{k}+(\xi_{i}+\xi_{j})x_{k}-\xi_{i}\pi_{j}}\in\mathbb{Q}[p_{k},x_{k}][[\pi_{i},\pi_{j},\xi_{i},\xi_{j}]]=\tilde{\mathcal{C}}(\\{i,j\\},\\{k\\})$

###### Proof.

The generating function $\mathcal{G}(m^{ij}_{k})$ is found by using Weyl’s
canonical commutation relation
$e^{\xi\mathbf{x}}e^{\pi\mathbf{p}}=e^{-\pi\xi}e^{\pi\mathbf{p}}e^{\xi\mathbf{x}}$
as follows.

$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ij}_{k}=m^{ij}_{k}(e^{\pi
p+\xi x})=e^{\pi p+\xi
x}//\mathbb{O}^{\otimes\\{i,j\\}}//\mathbf{m}^{ij}_{k}=$
$e^{\pi_{i}\mathbf{p}}e^{\xi_{i}\mathbf{x}}e^{\pi_{j}\mathbf{p}}e^{\xi_{j}\mathbf{x}}//(\mathbb{O}^{\otimes\\{k\\}})^{-1}=e^{(\pi_{i}+\pi_{j})\mathbf{p}}e^{-\xi_{i}\pi_{j}}e^{(\xi_{i}+\xi_{j})\mathbf{x}}//(\mathbb{O}^{\otimes\\{k\\}})^{-1}$
$=e^{(\pi_{i}+\pi_{j})p_{k}+(\xi_{i}+\xi_{j})x_{k}-\xi_{i}\pi_{j}}$

∎

Of course we already know that multiplication in $\mathcal{U}(\mathfrak{h})$
is associative but it is instructive to also check it using generating
functions. So instead of checking associativity directly in the form

$\mathbf{m}^{12}_{k}//\mathbf{m}^{k3}_{\ell}=\mathbf{m}^{23}_{k}//\mathbf{m}^{1k}_{\ell}$

we check the same equation after applying the functors
$\mathcal{O},\mathcal{G}$:

$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{12}_{k}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{k3}_{\ell}=\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{23}_{k}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{1k}_{\ell}$

The left hand side is computed using $\langle
f(s)e^{r\lambda}\rangle_{r,s}=f(\lambda)$ twice:

$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{12}_{k}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{k3}_{\ell}=\langle
e^{(\pi_{1}+\pi_{2})p_{k}+(\pi_{k}+\pi_{3})p_{\ell}+(\xi_{1}+\xi_{2})x_{k}+(\xi_{k}+\xi_{3})x_{\ell}-\xi_{1}\pi_{2}-\xi_{k}\pi_{3}}\rangle_{(\pi_{k},p_{k}),(\xi_{k},x_{k})}=$
$e^{(\pi_{1}+\pi_{2}+\pi_{3})p_{\ell}+(\xi_{1}+\xi_{2}+\xi_{3})x_{\ell}-\xi_{1}\pi_{2}-(\xi_{1}+\xi_{2})\pi_{3}}$

Notice that the generating function for multiplication is Gaussian, it is the
exponential of a quadratic. The composition of the generating functions was
rather easy because of the absence of terms $p_{i}x_{j}$ in the exponent. In
fact such terms $e^{px}$ cannot even occur in the ring
$\mathbb{Q}[p,x][[\pi,\xi]]$. On the other hand, in the next section we will
be interested in multiplying precisely such expressions. In knot theory they
are known as $R$-matrices and they are the elementary building blocks of the
computation, representing the crossings in the knot diagram. In this case the
form of the $R$-matrix and its inverse is

$\mathbf{R}_{ij}=e^{t(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbf{x}_{j}}\qquad\mathbf{R}^{-1}_{ij}=e^{-t(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbf{x}_{j}}$
(5)

Later we will see how these formulas naturally come out of the Drinfeld double
construction, see Section 5.3. The point of these elements is that they
provide solutions to the Yang-Baxter equation

$\mathbf{R}_{12}\mathbf{R}_{13}\mathbf{R}_{23}=\mathbf{R}_{23}\mathbf{R}_{13}\mathbf{R}_{12}$

that is central to both knot theory and integrable systems [4]. We will verify
this equation in the next section.

To accommodate the $R$-matrices we extend our algebra to an algebra over
$\mathbb{Q}[[t]]$. This can be done by simply tensoring all the constructions
we carried out so far with $\mathbb{Q}[[t]]$ and completing with respect to
the $t$-adic topology. We call this completion $\overline{U(\mathfrak{h})}$
and its elements are power series in $t$ with coefficients non-commutative
polynomials in $\mathbf{p}$ and $\mathbf{x}$. As such we have an isomorphism
of topological $\mathbb{Q}[[t]]$ modules
$\mathbb{Q}[p,x][[t]]\xrightarrow{\mathbb{O}}\overline{U(\mathfrak{h})}$ that
we call still $\mathbb{O}$. Provided we take $\mathrm{Hom}$ to refer to
$\mathbb{Q}[[t]]$ module maps there still is an isomorphism

$\mathrm{Hom}_{\mathbb{Q}[[t]]}(\mathbb{Q}[p_{J},x_{J}][[t]],\mathbb{Q}[p_{K},x_{K}][[t]])\cong\mathbb{Q}[p_{K},x_{K}][[t,\pi_{J},\xi_{J}]]$

Everything we said so far about generating functions continues to be true in
this setting. For a more thorough account of the infinite dimensional
intricacies involving topological $\mathbb{Q}[[t]]$ modules we refer to
Section 5.1. Here we will mostly gloss over the details of topological algebra
and move on to algebraic topology in the next section instead.

To compute with the $R$-matrices using the generating function
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ij}_{k}$ we need to map
it $\mathbf{R}_{ij}$ into an element
$R_{ij}=\mathbb{O}^{-1}(\mathbf{R}_{ij})\in\mathbb{Q}[p_{i},p_{j},x_{i},x_{j}][[t]]$.

######  Lemma 10.

$R_{ij}^{\pm 1}=(\mathbb{O}^{-1})^{\otimes\\{i,j\\}}(\mathbf{R}^{\pm
1}_{ij})=e^{(T^{\pm 1}-1)(p_{i}-p_{j})x_{j}}$ where $T=e^{-t}$.

###### Proof.

We will only prove the formula for $\mathbf{R}_{ij}$, leaving the similar
proof for $\mathbf{R}^{-1}_{ij}$ to the reader. Set $\Phi_{1}=\mathbf{R}_{ij}$
and $\Phi_{2}=\mathbb{O}(e^{(T-1)(p_{i}-p_{j})x_{j}})$ We will show
$\Phi_{1}=\Phi_{2}$ by proving both satisfy the same ODE in
$\overline{\mathbb{U}(\mathfrak{h})}^{\otimes\\{i,j\\}}$ given by
$\Phi|_{t=0}=1$ and
$\partial_{t}\Phi=(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbf{x}_{j}\Phi$.

The only non-trivial part to check is that $\Phi_{2}$ satisfies this ODE. Set
$\Psi=\mathbb{O}^{-1}(\Phi_{2})=e^{(T-1)(p_{i}-p_{j})x_{j}}$ and compute

$\partial_{t}\Phi_{2}=\mathbb{O}(\partial_{t}e^{(T-1)(p_{i}-p_{j})x_{j}})=\mathbb{O}(T(p_{i}-p_{j})x_{j}\Psi)=(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbb{O}(x_{j}\Psi+(T-1)x_{j}\Psi)$
$(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbb{O}(x_{j}\Psi-\partial_{p_{j}}\Psi)=(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbf{x}_{j}\mathbb{O}(\Psi)=(\mathbf{p}_{i}-\mathbf{p}_{j})\mathbf{x}_{j}\Phi_{2}$

The first equality sign comes from the formula
$\mathbf{x}f(\mathbf{p})=f(\mathbf{p})\mathbf{x}-\partial_{\mathbf{p}}f(\mathbf{p})$
for any power series $f$ in $\mathbf{p}$. ∎

As a first test let us multiply two $R$-matrices:
$\mathbf{F}=\mathbf{R}_{a1}\mathbf{R}_{b2}//\mathbf{m}^{ab}_{i}\in\overline{U(\mathfrak{h})}^{\otimes\\{i,1,2\\}}$.
For the knot theoretical interpretation of this computation see the next
section. In computations we prefer to work with
$F=\mathbb{O}^{-1}(\mathbf{F})$ and use generating functions:

$F=R_{a1}R_{b2}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ab}_{i}$
(6)

First, Lemmas 10 and 9
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{ij}_{k}$ tell us that

$F=\langle
e^{(T-1)\big{(}(p_{a}-p_{1})x_{1}+(p_{b}-p_{2})x_{2}\big{)}-\xi_{a}\pi_{b}+(\xi_{a}+\xi_{b})x_{0}+(\pi_{a}+\pi_{b})p_{0}}\rangle_{a,b}=\langle
e^{c+rWs+gs+rf}\rangle_{a,b}$

Here we abbreviated the contraction notation to just state that all the pairs
$p_{u},\pi_{u}$ and $x_{u},\xi_{u}$ should be contracted for $u\in\\{a,b\\}$.
Also in the final formula we have
$r=(p_{a},p_{b},\xi_{a},\xi_{b}),s=(\pi_{a},\pi_{b},x_{a},x_{b})$,
$c=(1-T)(p_{1}x_{1}+p_{2}x_{2})$ and $W=-E^{3}_{2}$, where $E^{i}_{j}$ denotes
the elementary matrix with $1$ at the $(i,j)$-th entry and zero elsewhere.
Finally $f=((T-1)x_{1},(T-1)x_{2},x_{0},x_{0})$ and $g=(p_{0},p_{0},0,0)$.
Since $\tilde{W}=I-E^{3}_{2}$ the Contraction lemma implies that

$F=\det(\tilde{W})e^{c+g\tilde{W}f}=e^{(T-1)\big{(}(p_{i}-p_{1})x_{1}+(p_{i}-p_{2})x_{2}\big{)}}$

Calculations such as the above may be simplified by dealing with uncomplicated
contractions using an easy case of formula (4):

######  Lemma 11.

If $G=ux_{i}+vp_{j}$ and $u,v$ do not depend on either of $p_{j}$ or $x_{i}$
then

$e^{G}//m^{ij}_{k}=e^{G-uv}|_{i,j\mapsto k}$ (7)

###### Proof.

By definition

$e^{ux_{i}+vp_{j}}//m^{ij}_{k}=\langle
e^{ux_{i}+vp_{j}+(\pi_{i}+\pi_{j})p_{k}+(\xi_{i}+\xi_{j})x_{k}-\xi_{i}\pi_{j}}\rangle_{i,j}$

To contract this formula we repeatedly use $\langle f(r)e^{\lambda
s}\rangle_{r,s}=f(\lambda)$. Contracting the pairs $\pi_{i},p_{i}$ and
$\xi_{j},x_{j}$ yields

$\langle
e^{ux_{i}+vp_{j}+\pi_{j}p_{k}+\xi_{i}x_{k}-\xi_{i}\pi_{j}}\rangle_{x_{i},p_{j}}|_{x_{j}\mapsto
x_{k},p_{i}\mapsto p_{k}}$

Next, contracting $\xi_{i},x_{i}$ replaces $x_{i}\mapsto x_{k}-\pi_{j}$ and
deletes the $\xi_{i}$ terms:

$\langle
e^{u(x_{k}-\pi_{j})+vp_{j}+\pi_{j}p_{k}}\rangle_{p_{j}}|_{x_{j}\mapsto
x_{k},p_{i}\mapsto p_{k}}$

Finally we replace $p_{j}$ by $p_{k}-u$ to obtain

$e^{ux_{k}+vp_{k}-uv}|_{x_{j}\mapsto x_{k},p_{i}\mapsto p_{k}}$

∎

Applying formula (7) to $e^{G}$ with
$G=(T-1)\big{(}(p_{1}-p_{i})x_{i}+(p_{2}-p_{j})x_{j}\big{)}$ we see that

$e^{G}//m^{12}_{0}=e^{G}|_{1,2\mapsto
0}=e^{(T-1)\big{(}(p_{0}-p_{i})x_{i}+(p_{0}-p_{j})x_{j}\big{)}}$

since our $G$ does not even depend on $x_{1}$.

We close this section with an example where a denominator does arise:

$R_{12}//m^{21}_{0}=\langle
e^{(T-1)(p_{1}-p_{2})x_{2}+(\pi_{1}+\pi_{2})p_{0}+(\xi_{1}+\xi_{2})x_{0}-\xi_{2}\pi_{1}}\rangle_{1,2}$

Choosing $r=(p_{1},p_{2},\xi_{1},\xi_{2})$ and
$s=(\pi_{1},\pi_{2},x_{1},x_{2})$ we may write the exponent as $rWs+rf+gs$
with

$1-W=\left(\begin{array}[]{cccc}1&0&0&1-T\\\ 0&1&0&T-1\\\ 0&0&1&0\\\
1&0&0&1\end{array}\right)\quad f=(0,0,x_{0},x_{0})\quad g=(p_{0},p_{0},0,0)$

By the contraction Lemma 5 we find $R_{12}//m^{21}_{0}=\frac{1}{T}$. In the
next section we will see what this has to do with the Reidemeister I move in
knot theory.

##  4 From algebras to tangle invariants 1

###  4.1 Tangle diagrams 1

This section is intended to showcase the application of generating functions
to knot and tangle invariants in a simplified and less technical case. We
simplified matters in two ways: first the algebra used is just the completed
Heisenberg algebra $\overline{U(\mathfrak{h})}$ introduced in the previous
section. Second, the notion of tangle diagram used here is more or less
standard. It does not take into account rotation numbers as Morse diagrams
would. This section may be skipped without loss of continuity in the rest of
the paper. We nevertheless hope that studying this simplified case will help
understanding our way of thinking.

Figure 1: Left: A tangle diagram with three underlying disks (in grey) and
eight strands. Right: a tangle diagram with one strand and one underlying disk
that respresents the knot $8_{17}$.

######  Definition 12.

(Tangle diagrams)
A tangle diagram $D$ consists of a finite union of disjoint closed topological
disks in the plane, called the underlying disks, notation: $|D|$. In each disk
at least one closed interval is properly immersed. The immersed intervals
(also known as strands) intersect transversally in what we refer to as
crossings. Each crossing comes with a sign $\pm$ that is indicated as in
Figure 2 (left). Finally our strands are labelled with distinct elements of
some set, oriented and have distinct endpoints on the boundary of the disk.

In Figure 1 a typical tangle diagram $D$ is shown on the left with $|D|$
consisting of three disks and $8$ strands. We often think about tangles as
pieces of a knot and in this example the relevant knot is $8_{17}$ shown in
the same figure as a long knot (one strand tangle). The reader is warned that
our tangle diagrams are slightly non-standard in that like string links they
do not have closed components however unlike string links they are not
embedded in a single disk and their endpoints can be anywhere on the boundary.
As we will see this type of tangles is well suited for the universal knot
invariants we are about to introduce.

The simplest tangle diagrams are the crossingless diagrams and the two
crossings, see Figure 2. A single strand in a single disk without crossings is
called $1_{i}$. The positive and negative crossings where the two strands are
labeled $i$ and $j$ and we always list the over-passing strand label first are
called $X_{ij}$ and $X_{ij}^{-1}$. When convenient we also use the alternative
notation $X_{ij}^{-1}=\bar{X}_{ij}$.

Figure 2: The simplest oriented tangles. Left: the positive and negative
crossing. Middle: some crossingless tangles. Right: the three strand tangle
$F$.

We often think of tangle diagrams as being assembled from crossings using two
operations called merging and disjoint union.

######  Definition 13.

(Disjoint union and merging of tangle diagrams)
Given two tangle diagrams $D,E$ we define their disjoint union (notation:
$DE$) to be their union after applying a planar isotopy to make sure their
underlying disks $|D|$ and $|E|$ are disjoint.

The merge $E=D//m^{ij}_{k}$ of diagram $D$ connects the endpoint of strand $i$
with the beginning of strand $j\neq i$ by an embedded interval $c$ in the
plane disjoint from the disks of $D$. $|E|$ is obtained from $|D|$ by taking
the union with a tubular neighborhood $B$ of $c$ that does not intersect more
of $|E|$ and its strands than necessary444this means $B$ is a band two of
whose opposite sides are attached to sufficiently small intervals in the
boundary of the disk(s) containing the endpoint of $i$ and the start point of
$j$ that does not meet any other strands. as shown in Figure 3 below. In case
an annulus is created by attaching $B$ the operation is only allowed if one of
the boundary components of the annulus does not contain any ends of strands.
In that case a disk will be attached to turn $|E|$ into a union of disks once
more.

Figure 3: Merging a pair of strands in tangle diagram $D$ (3rd picture).
Either two underlying disks become one (Right) or an annulus is created (2nd
picture) that is then capped off (Left).

Even though the merging operation $m^{ij}_{k}$ may not always be defined, it
does allow us to construct any tangle diagram from a disjoint union of
crossings555Technically the crossingless diagrams cannot be constructed this
way.. Just start with the desired diagram, cut the crossings loose and then
merge them back. A concrete example of this process is shown in 5.
Alternatively we could write this trefoil diagram as

$\mathcal{T}=X_{12}X_{34}X_{56}//m^{14}_{a}//m^{23}_{b}//m^{a5}_{i}//m^{b6}_{j}//m^{ij}_{0}$

Tangle diagrams are meant to represent tangles. Concretely, a diagram $D$ with
underlying disks $|D|$ is to be interpreted as intervals properly embedded
into $|D|\times[-1,1]$ with $|D|$ the underlying disks of our diagram and the
endpoints all distinct on $|D|\times\\{0\\}$. Such embeddings are to be taken
up to isotopy fixing the endpoints. By a straightforward extension of the
classical Reidemeister theorem this gives rise to the following notion of
equivalence of tangle diagrams.

######  Definition 14.

(Equivalence of diagrams)
Generate an equivalence relation $"="$ on tangle diagrams by the following
rules, where $D,E,F$ are tangle diagrams:

  1. 1.

$D=E$ if $D,E$ are planar isotopic respecting the orientation and labels on
the strands.

  2. 2.

If $D=E$ then $DF=EF$.

  3. 3.

If $D=E$ then $D//m^{ij}_{k}=E//m^{ij}_{k}$, provided both make sense.

  4. 4.

$D=E$ if $D$ and $E$ appear in one of the Reidemeister equalities shown in
Figure 4.

Figure 4: The oriented Reidemeister moves, together with their algebraic
description.

In using the notation $=$ for the equivalence relation between tangle diagrams
we abuse our notation slightly in that strict equality is not expressible
anymore but this will not cause any problems in the sequel.

As hinted at above we assert that isotopy classes of tangles in a disjoint
union of cylinders are in bijection with equivalence classes of diagrams. Also
tangles with a single strand coincide with long knots which are well known to
be equivalent to round knots666embeddings of the circle, see for example a
diagram of the knot $8_{17}$ on the right of Figure 1 and a long trefoil in
Figure 5 below.

As shown in Figure 4 the Reidemeister equivalences can be written
algebraically as follows.

$\displaystyle X_{12}//m^{21}_{i}=1_{i}=X_{12}//m^{12}_{i}$ (8) $\displaystyle
X^{-1}_{12}X_{34}//m^{13}_{i}//m^{24}_{j}=1_{ij}\qquad
X^{-1}_{12}X_{34}//m^{13}_{i}//m^{42}_{j}=1_{i\bar{j}}$ (9) $\displaystyle
F_{i12}X_{34}//m^{13}_{j}//m^{24}_{k}=F_{i12}X_{34}//m^{32}_{j}//m^{41}_{k}\qquad
F_{i12}=X_{a1}X_{b2}//m^{ab}_{i}$ (10)

###  4.2 Universal tangle invariants 1

The tangle invariants we are about to introduce are defined by specifying
their value on the crossings and giving rules for how they behave under
disjoint union and merging. In this section we use the algebra developed so
far to provide invariants $\tilde{\mathbf{Z}}_{A}$ of knots and tangles. This
invariant is a simplified version of the so called universal knot invariant
first introduced by Lawrence [10], see also [14]. This is a simplified version
of the full-fledged universal invariant $Z$ treated in Section 7. We also use
a simplified model of tangle diagrams here as described in the previous
subsection.

######  Definition 15.

Suppose $A$ is an associative algebra $A$ with unit $\mathbf{1}$ and
multiplication $\mathbf{m}^{ij}_{k}:A^{\otimes\\{i,j\\}}\to
A^{\otimes\\{k\\}}$ and we have chosen elements $\mathbf{R}^{\pm 1}_{ij}\in
A^{\otimes\\{i,j\\}}$. For a tangle diagram $D$ whose strands are labeled by
set $L$ define $\tilde{\mathbf{Z}}_{A}(D)\in A^{\otimes L}$ by the following
rules.

  1. 1.

If $D$ does not have crossings, $\tilde{\mathbf{Z}}_{A}(D)=1^{\otimes L}$.
(crossingless diagrams don’t count).

  2. 2.

$\tilde{\mathbf{Z}}_{A}(X_{ij}^{\pm 1})=\mathbf{R}^{\pm 1}_{ij}$ (value of the
crossings).

  3. 3.

If diagram $E$ is labelled by set $M$ then
$\tilde{\mathbf{Z}}_{A}(DE)=\tilde{\mathbf{Z}}_{A}(D)\otimes\tilde{\mathbf{Z}}_{A}(E)\in
A^{\otimes L\sqcup M}$
(disjoint union is tensor product).

  4. 4.

$\tilde{\mathbf{Z}}_{A}(D//m^{ij}_{k})=\tilde{\mathbf{Z}}_{A}(D)//\mathbf{m}^{ij}_{k}$
(merging is multiplication).

We will often use the notation
$\tilde{Z}=\mathbb{O}^{-1}(\tilde{\mathbf{Z}})$.

In the literature the universal invariants are usually presented more
informally as follows. Place a copy of $\mathbf{R}$ on each positive crossing
of the diagram and a copy of $\mathbf{R}^{-1}$ on each negative crossing so
that the first tensor factor is assigned to the over-strand and the second
tensor factor to the under strand. For every strand we multiply the elements
of $A$ in order of appearance and finally we tensor over all strand labels.
This description matches ours as can be seen by induction on the number of
crossings and merges necessary to construct the tangle diagram.

Associativity of the algebra will make sure the above rules do in fact define
a value $\tilde{\mathbf{Z}}_{A}(D)$ on each of our tangle diagrams. However
without further assumptions equivalent diagrams may be assigned completely
different values. To ensure $\tilde{\mathbf{Z}}_{A}(D)$ is a true tangle
invariant it suffices to ensure that $\tilde{\mathbf{Z}}_{A}$ takes the same
value on each of the Reidemeister tangle equalities (8). Applying the defining
rules for $\tilde{\mathbf{Z}}_{A}$ to each side transforms these equations
into equations for the algebra $A$ and the chosen $\mathbf{R}^{\pm 1}_{ij}$
(traditionally known as $R$-matrices).

$\displaystyle\mathbf{R}_{12}//\mathbf{m}^{21}_{i}=1_{i}=\mathbf{R}_{12}//\mathbf{m}^{12}_{i}$
(11)
$\displaystyle\mathbf{R}^{-1}_{12}\mathbf{R}_{34}//\mathbf{m}^{13}_{1}//\mathbf{m}^{24}_{2}=\mathbf{1}_{12}\qquad\mathbf{R}^{-1}_{12}\mathbf{R}_{34}//\mathbf{m}^{13}_{1}//\mathbf{m}^{42}_{2}=\mathbf{1}_{1\bar{2}}$
(12)
$\displaystyle\tilde{\mathbf{Z}}_{A}(F_{i12})\mathbf{R}_{34}//\mathbf{m}^{13}_{j}//\mathbf{m}^{24}_{k}=\tilde{\mathbf{Z}}_{A}(F_{i12})\mathbf{R}_{34}//\mathbf{m}^{32}_{j}//\mathbf{m}^{41}_{k}\qquad\tilde{\mathbf{Z}}_{A}(F_{i12})=\mathbf{R}_{a1}\mathbf{R}_{b2}//\mathbf{m}^{ab}_{i}$
(13)

In what follows we will illustrate the invariant concretely by working with
the Heisenberg algebra. So throughtout the section we will assume
$A=\overline{U(\mathfrak{h})}$ and will abbreviate
$\tilde{\mathbf{Z}}=\tilde{\mathbf{Z}}_{\overline{U(\mathfrak{h})}}$ and
$\tilde{Z}=\mathbb{O}^{-1}\tilde{\mathbf{Z}}$. In this case we can illustrate
the power of the Contraction Lemma 6. For convenience we work with generating
functions throughout and recall (Lemma 10) the $R$-matrices of
$\overline{U(\mathfrak{h})}$ are

$R_{ij}^{\pm 1}=e^{(T^{\pm
1}-1)(p_{i}-p_{j})x_{j}}=\mathbb{O}^{-1}(\mathbf{R}^{\pm}_{ij})$

We are now in a position to verify the Reidemeister moves explicitly in the
form (11). This will show that $\tilde{\mathbf{Z}}$ is well-defined up to
multiplication by a power of $\pm T$ where $T=e^{t}$. The ambiguity in powers
of $T$ comes from the fact that the Reidemeister I move is only satisfied up
to such powers. Indeed at the end of the previous section we already checked
that $R_{12}//m^{21}_{0}=T^{-1}$. We leave Reidemeister II to the reader and
spend the remainder of this section verifying Reidemeister III and computing
the value of the trefoil knot in Figure 5.

We already computed the invariant of the three strand tangle $F_{i12}$ shown
to the right in Figure 2. We read Reidemeister III as the act of sliding a
crossing through $F$ and so it is useful to recall that we computed in the
previous section (see Equation (6)) that

$F_{i12}=e^{(T-1)\big{(}(p_{i}-p_{1})x_{1}+(p_{i}-p_{2})x_{2}\big{)}}$

The left hand side of Reidemeister III (Equation (13)) after applying our
functors $\mathcal{O}$ and $\mathcal{G}$ becomes

$LHS=F_{i12}R_{34}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{13}_{j}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{24}_{k}$

We compute this step by step using $e^{G}//m^{ij}_{k}=e^{G-uv}|_{i,j\mapsto
k}$ where $G=ux_{i}+vp_{j}$ is a quadratic with no factor $x_{i}p_{j}$, see
Lemma 11:

$LHS=e^{G}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{13}_{j}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{24}_{k}=e^{G^{\prime}}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{24}_{k}=e^{(T-1)\big{(}(p_{i}-p_{j})x_{j}+(p_{j}-p_{k})x_{k}-(T-1)(p_{i}-p_{j})x_{k}+T(p_{i}-p_{k})x_{k}\big{)}}$

Here we set
$G=(T-1)\big{(}(p_{i}-p_{1})x_{1}+(p_{i}-p_{2})x_{2}+(p_{3}-p_{4})x_{4}\big{)}$
and in the first step we subtract $(T-1)^{2}(p_{i}-p_{1})x_{4}$ and substitute
$1,3\mapsto j$ to get
$G^{\prime}=(T-1)\big{(}(p_{i}-p_{j})x_{j}+(p_{i}-p_{2})x_{2}+(p_{j}-p_{4})x_{4}-(T-1)(p_{i}-p_{j})x_{4}\big{)}$.
In the final step we subtract $-(T-1)^{2}(p_{i}-p_{2})x_{4}$ and substitute
$2,4\mapsto k$ to obtain the formula for LHS shown above.

Similarly the right hand side
$RHS=F_{i12}R_{34}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{32}_{j}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{41}_{k}$
is computed in two steps as

$RHS=e^{G}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{32}_{j}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{41}_{k}=e^{G^{\prime\prime}}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m^{41}_{k}=e^{(T-1)\big{(}(p_{i}-p_{k})x_{k}+(p_{i}-p_{j})x_{j}+(p_{j}-p_{k})x_{k}+(T-1)(p_{j}-p_{k})x_{k}\big{)}}$

where $G$ is as above and not depend on $x_{3}$ so that we get
$G^{\prime\prime}$ by simply renaming $3,2\mapsto j$:
$G^{\prime\prime}=e^{(T-1)\big{(}(p_{i}-p_{1})x_{1}+(p_{i}-p_{j})x_{j}+(p_{j}-p_{4})x_{4}\big{)}}$.
In the final step we subtract $-(T-1)^{2}(p_{j}-p_{4})x_{1}$ and substitute
$4,1\mapsto k$ to get the formula shown. This verifies Reidemeister III.

Figure 5: Left to right: Building up a diagram of a long trefoil knot by
merging the disjoint union of three crossings.

As another example let us compute the invariant on the (long, positive)
trefoil knot shown in Figure 5 (right). Denote by $\mathcal{D}_{n}$ then
$2$-strand braid diagram with $n$ positive crossings and strands named $i,j$,
where $i$ is the final overpass. Notice that $\mathcal{D}_{1}=X_{ij}$ and
$\mathcal{D}_{2}=X_{12}X_{34}//m^{23}_{i}//m^{14}_{j}$ and
$\mathcal{D}_{3}=\mathcal{D}_{2}X_{12}//m^{1i}_{i}//m^{2j}_{j}$ as shown in
Figure 5. After computing the invariant on these diagrams we compute the
invariant of the trefoil using $\mathcal{T}=\mathcal{D}_{3}//m^{ij}_{0}$.

Using Lemma 11 as in the Reidemeister III example above we compute

$\tilde{\mathbf{Z}}(\mathcal{D}_{2})=R_{12}R_{34}//m^{14}_{j}//m^{23}_{i}=e^{(T-1)(p_{i}-p_{j})(x_{i}-Tx_{j})}$

Next, to find $\mathcal{D}_{3}$ we merge another crossing:
$\tilde{\mathbf{Z}}(\mathcal{D}_{3})=\tilde{\mathbf{Z}}(\mathcal{D}_{2})R_{12}//m^{1i}_{i}//m^{2j}_{j}=$

$e^{(T-1)\big{(}(p_{i}-p_{j})(x_{i}-Tx_{j})+(p_{1}-p_{2})x_{2}\big{)}}//m^{1i}_{i}//m^{2j}_{j}=e^{(T-1)(p_{i}-p_{j})\big{(}-Tx_{i}+(1+T^{2})x_{j}\big{)}}$

To complete the trefoil computation we need the full power of the contraction
Lemma 5
$\tilde{\mathbf{Z}}(\mathcal{T})=\tilde{\mathbf{Z}}(\mathcal{D}_{3})//m^{ij}_{0}=$

$e^{(T-1)(p_{i}-p_{j})\big{(}-Tx_{i}+(1+T^{2})x_{j}\big{)}+(\pi_{i}+\pi_{j})p_{0}+(\xi_{i}+\xi_{j})x_{0}-\xi_{i}\pi_{j}}//m^{ij}_{0}=\langle
e^{rf+gs+rWs}\rangle_{i,j}=\frac{1}{1-T+T^{2}}$

using $r=(p_{i},p_{j},\xi_{i},\xi_{j})$, $s=(\pi_{i},\pi_{j},x_{i},x_{j})$ and
with $\tilde{W}=(1-W)^{-1}$:

$W=\left(\begin{array}[]{cccc}0&0&-(T-1)T&-(1-T)\left(T^{2}+1\right)\\\
0&0&-(1-T)T&-(T-1)\left(T^{2}+1\right)\\\ 0&-1&0&0\\\ 0&0&0&0\\\
\end{array}\right)$

and $f=(0,0,x_{0},x_{0})$ and $g=(p_{0},p_{0},0,0)$. Recall
$\tilde{W}=(1-W)^{-1}$ and that according to the contraction lemma the
contraction should be $e^{g\tilde{W}f}\det\tilde{W}=\frac{1}{1-T+T^{2}}$
because the exponent $g\tilde{W}f$ is zero.

Our conclusion then is that

$\tilde{\mathbf{Z}}(\mathcal{T})=\frac{1}{1-T+T^{2}}$

This concludes our warmup on the Heisenberg algebra invariant. We will leave
it to the reader to prove one always finds the reciprocal of the Alexander
polynomial of a knot. A similar result will be proven as the $\epsilon=0$ part
of Theorem 51.

###  4.3 Computer practicum 1

The reader has noticed that although not particularly hard, the computation of
$\tilde{\mathbf{Z}}$ takes some effort to do by hand even for simple tangles.
Using the more general Mathematica program explained in Appendix B, the
invariant $\tilde{\mathbf{Z}}$ can be computed quickly as follows. First we
set the parameters $\$k$ and $\hbar$ that do not play a role now will be
important in later sections.

The notation used in the program is similar to the one in the main text except
for the following details. We use $y$ and $\eta$ instead of $p$ and $\pi$.
Also A morphism in $\tilde{\mathcal{C}}(J,K)$ of the form $Pe^{G}$ is denoted
by $\mathbb{E}_{J\to K}[G,P]$, where $G$ is supposed to be a quadratic in
$x,y,\xi,\eta$ with coefficients in $\mathbb{Q}(T)$ and $P$ is also a rational
function of $T=e^{-t}$. When $P=1$ it is omitted from the $\mathbb{E}$
notation. The program also uses $hm_{i,j\to k}$ for our $m^{ij}_{k}$ and
$hR_{ij}$ for $R_{ij}$. As a final touch there is a command for replacing
$e^{-t}$ by $T$ and this is called by appending /. l2U. With this notation in
place we can easily recompute the results we just did by hand and many more.
Our convention is to write the input in boldface and the output right below
it. Sometimes we string together several input lines in a single list. This
has the advantage that the output will also be a single list of outputs.

First we investigate invariance under the Reidemeister moves. We see that
Reidemeister 1 fails as expected and Reidemeister 2 holds. Reidemeister 3 is
checked as in the main text by first computing the value of the tangle F. We
then checked equality (using $\equiv$) between the two sides of the
Reidemeister move obtaining the reassuring output True

Notice how after applying the Simplify and l2U command we do indeed find the
same expression for the three strand tangle F as we did before.

Next we check again the computation for $D_{2},D_{3}$ leading up to the value
of the trefoil:

As a bonus we compute the universal invariant for the knot $8_{17}$ using a Do
loop:

The above description of the knot as a merging of eight crossings can be read
off directly from Figure 1. One just enumerates the over- and underpasses as
one walks along the strand and uses these labels for the eight crossings
involved.

##  5 Main example

We introduce a Hopf algebra $\mathbb{B}$ that forms the basis for our main
example $\mathbb{D}$. The algebra $\mathbb{B}$ is based on the two-dimensional
non-commutative Lie algebra so in some sense the simplest possible algebra of
its sort. Next we construct the Drinfeld double $\mathbb{D}$ of our algebra
$\mathbb{B}$. The point of the double construction is that $\mathbb{D}$
automatically comes with a solution to the Yang-Baxter equation called the
(universal) $R$-matrix. Roughly speaking
$\mathbb{D}=\mathbb{B}\otimes\mathbb{B}^{*}$ as a co-algebra but with a
twisted product explained below. The product on $\mathbb{D}$ is designed so
that $\mathbb{D}$ becomes a quasi-triangular Hopf algebra. Even better, the
$R$-matrix is just the identity $\mathrm{id}_{\mathbb{B}}$! (viewed as an
element in $\mathbb{B}\otimes\mathbb{B}^{*}$). In our case we are even more
lucky and $\mathbb{D}$ contains a ribbon element. All these notions are
crucial to constructing the universal knot and tangle invariants in Section 7
refining the invariants $\tilde{\mathbf{Z}}$ from Section 4.

###  5.1 Two-dimensional algebra and two-step Gaussians

The foundation for our main example is the algebra $\mathbb{B}$ for beginning
or Borel. The latter refers to the close connection to the Borel part of
$\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$. The reader is warned in advance that
our variable $\epsilon$ is not playing the role of the usual $\hbar$ in
quantized enveloping algebras of Drinfeld-Jimbo type [5]. One key difference
is that $\epsilon$ does not appear in the co-product of Definition 17 below.

As the dimension of our algebras is infinite we need to take some care to
define a topology to make sense of infinite series such as $e^{b\hbar}$. We
generally use formal power series in $\hbar$ and the $\hbar$-adic topology as
recalled below.

######  Definition 16.

($\hbar$-adic topology)

  1. 1.

Define $\mathbb{Q}_{\hbar}[z]=\mathbb{Q}[z][[\hbar]]$ and set
$\mathbb{K}=\mathbb{Q}_{\hbar}[\epsilon]$.

  2. 2.

Recall the $\hbar$-adic norm on $\mathbb{Q}_{\hbar}[z]$ is defined by
$|f|=2^{-k}$ if $\hbar^{k}$ is the highest power of $\hbar$ dividing $f$ (and
$|0|=0$ by convention).

  3. 3.

We say a topological $\mathbb{K}$-module $M$ is topologically generated by
generators $\mathbf{z}$ if all $m\in M$ can be written as a (convergent)
series in $\hbar$ whose coefficients are non-commutative polynomials in
$\mathbf{z}$ with coefficients in $\mathbb{Q}[\epsilon]$.

  4. 4.

Finally $M$ is of PBW-type if there exists an isomorphism of topological
$\mathbb{K}$-modules $\mathbb{O}:\mathbb{Q}_{\hbar}[\epsilon,z]\to M$.

In what follows, tensor products will always be taken over $\mathbb{K}$ and it
is understood that the tensor product will be completed. For the PBW-type
modules we work with the completion process is relatively straightforward in
that if the modules are isomorphic to $\mathbb{Q}_{\hbar}[a]$ and
$\mathbb{Q}_{\hbar}[b]$ then the completed tensor product will be isomorphic
to $\mathbb{Q}_{\hbar}[a,b]$. A relevant example of why this is necessary is
to deal with expressions like
$e^{a_{1}a_{2}\hbar}\in\mathbb{Q}_{\hbar}[a]^{\otimes 2}$.

Notice that $\mathbb{K}$-module maps between topologically generated
$\mathbb{K}$ modules are automatically continuous with respect to the chosen
topologies. As such they are completely determined by the images of the finite
monomials. For a more complete exposition on topological algebras in the
context of quantum groups we refer to [8] chapter XVI. The casual reader can
mostly ignore the topological subtleties.

######  Definition 17.

(The Hopf algebra $\mathbb{B}$)
The topological $\mathbb{K}$-module $\mathbb{B}$ is topologically generated by
$\mathbf{y}$ and $\mathbf{b}$ subject to the relation

$[\mathbf{y},\mathbf{b}]=\epsilon\mathbf{y}$

Also define $\mathbb{O}:\mathbb{Q}_{\hbar}[\epsilon,y,b]\to\mathbb{B}$ by

$\mathbb{O}\big{(}\sum_{\ell}\sum_{i,j}f_{ij}y^{j}b^{i}\hbar^{\ell}\big{)}=\sum_{\ell}\sum_{i,j}f_{ij}\mathbf{y}^{j}\mathbf{b}^{i}\hbar^{\ell}$

Setting $\mathbf{B}=e^{-\hbar\mathbf{b}}$ and $q=e^{\epsilon\hbar}$ we define
a (topological) Hopf algebra structure by

$\mathbf{\Delta}(\mathbf{y})=\mathbf{y}_{2}+\mathbf{y}_{1}\mathbf{B}_{2}\quad\mathbf{\Delta}(\mathbf{b})=\mathbf{b}_{1}+\mathbf{b}_{2}\qquad\mathbf{S}(\mathbf{y})=-\mathbf{y}\mathbf{B}^{-1}\quad\mathbf{S}(\mathbf{b})=-\mathbf{b}$

Finally the co-unit777Not to be confused with our deformation parameter
$\epsilon$. $\varepsilon$ sends both $\mathbf{y},\mathbf{b}$ to $0$.

It should be clear that $\mathbb{O}$ makes $\mathbb{B}$ of PBW type where we
ordered the monomials anti-alphabetically $y,b$ in honor of Yang-Baxter. The
reader is invited to verify that extending the co-unit and co-product
multiplicatively and the antipode anti-multiplicatively this does indeed
define a Hopf algebra. We remark that $\mathbb{B}$ is the simplest instance of
at least two constructions of Hopf algebras. For example $\mathbb{B}$ can be
understood as the bosonisation of the braided line of Majid (Thm 16.4 [13]).
$\mathbb{B}$ may also be viewed as a quantization of the two-dimensional Lie
bi-algebra with generators $y,b$ and bracket as shown and ”standard” co-
bracket $\delta(b)=0$, $\delta(y)=y\wedge b$ (see sec 6.4 of [4]). The form of
the co-product is in fact dual to that of the product making, $\mathbb{B}$
self-dual in some sense. In the next section we will give a precise meaning to
this statement.

Before moving on to the main example in the next section we briefly explore
what generating functions for the Hopf algebra operations of $\mathbb{B}$ look
like. Many of the complications we will meet later find their origin here.

Each of the operations will be viewed as a $\mathbb{K}$ module map
$\mathbf{f}:\mathbb{B}^{\otimes J}\to\mathbb{B}^{\otimes K}$. Using the
notation introduced in Section 2 but working over $\mathbb{K}$ instead of
$\mathbb{Q}$, the corresponding $\mathbb{K}$-module map is:

$\mathcal{O}(\mathbf{f})=\mathbb{O}//\mathbf{f}//\mathbb{O}^{-1}=f\in\mathrm{Hom}(\mathbb{Q}_{\hbar}[\epsilon,y_{J},b_{J}],\mathbb{Q}_{\hbar}[\epsilon,y_{K},b_{K}])$

We then compute the generating function
$\mathcal{G}(f)\in\mathbb{Q}[\epsilon,y_{K},b_{K}][[\hbar,\eta_{J},\beta_{J}]]$
by the formula $\mathcal{G}(f)=f(e^{\beta_{J}b_{J}+\eta_{J}y_{J}})$. Recall
that $\mathcal{G}(\mathrm{id}_{J})=e^{\beta_{J}b_{J}+\eta_{J}y_{J}}$ and this
is just a fast way to write the usual exponential generating function.

######  Lemma 18.

(Generating multiplication in $\mathbb{B}$)

$\mathcal{G}(m^{ij}_{k})=e^{(\beta_{i}+\beta_{j})b_{k}+(e^{-\epsilon\beta_{i}}\eta_{i}+\eta_{j})y_{k}}$

###### Proof.

Since $e^{\beta\mathbf{b}}\mathbf{y}=\mathbf{y}e^{\beta(\mathbf{b}-\epsilon)}$
we have

$e^{\beta\mathbf{b}}e^{\eta\mathbf{y}}=e^{\beta\mathbf{b}}\sum_{k}\frac{\eta^{k}}{k!}\mathbf{y}^{k}=\sum_{k}\frac{\eta^{k}}{k!}\mathbf{y}^{k}e^{-k\epsilon\beta}e^{\beta\mathbf{b}}=e^{e^{-\epsilon\beta}\eta\mathbf{y}}e^{\beta\mathbf{b}}$

It follows that
$\mathcal{G}(m^{ij}_{k})=e^{\beta_{i}b_{i}+\beta_{j}b_{j}+\eta_{i}y_{i}+\eta_{j}y_{j}}//\mathbb{O}//\mathbf{m}^{ij}_{k}//\mathbb{O}^{-1}=$

$e^{\eta_{i}\mathbf{y}_{k}}e^{\beta_{i}\mathbf{b}_{k}}e^{\eta_{j}\mathbf{y}_{k}}e^{\beta_{j}\mathbf{b}_{k}}//\mathbb{O}^{-1}=e^{\eta_{i}\mathbf{y}_{k}+e^{-\epsilon\beta_{i}}\eta_{j}\mathbf{y}_{k}}e^{\beta_{i}\mathbf{b}_{k}+\beta_{j}\mathbf{b}_{k}}//\mathbb{O}^{-1}=e^{(\beta_{i}+\beta_{j})b_{k}+(e^{-\epsilon\beta_{i}}\eta_{i}+\eta_{j})y_{k}}$

∎

When $\epsilon=0$ this generating function is of the same Gaussian form as the
one we found in Section 2 for the commutative multiplication of polynomials
$\mathbb{Q}[y,b]$. This is correct because the commutation relation is
$[\mathbf{y},\mathbf{b}]=\epsilon\mathbf{y}$. However, once $\epsilon$ is non-
zero the generating function $\mathcal{G}(m^{ij}_{k})$ is no longer of the
Gaussian type that we can compose directly using the Contraction Theorem 6.

Fortunately it is a perturbed Gaussian in _two_ different senses: 1) as seen
from the point of view of $y$ or 2) from the perspective of $b$. Indeed if we
regard $y,\eta$ as constants then $\mathcal{G}(m^{ij}_{k})$ is a perturbed
Gaussian in $\beta,b$ with perturbation
$e^{(e^{-\epsilon\beta_{j}}\eta_{i}+\eta_{j})y_{k}}$. Conversely, if we fix
$\beta,b$ then $\mathcal{G}(m^{ij}_{k})$ is a Gaussian in $\eta,y$ with
constant perturbation. The upshot is that we can still multiply in
$\mathbb{B}$ using the contraction theorem but we have to do carry out the
multiplication in two steps: first contract $\beta,b$ and then $y,\eta$ (or
the other way around). We call perturbed Gaussian expressions of this type
two-step Gaussians.

The generating function for the co-product is more challenging and this is one
reason why we introduced $\epsilon$. The generating function is

$\mathcal{G}(\Delta^{i}_{jk})=\mathbb{O}^{-1}\mathbf{\Delta}^{i}_{jk}(e^{\eta_{i}\mathbf{y}_{i}}e^{\beta_{i}\mathbf{b}_{i}})=\mathbb{O}^{-1}\big{(}\mathbf{\Delta}(e^{\eta_{i}\mathbf{y}_{i}})\mathbf{\Delta}(e^{\beta_{i}\mathbf{b}_{i}})\big{)}$

and the problem is that the factor depending on $\mathbf{y}$ does not seem to
have a Gaussian expression. Such co-products are often expanded in terms of
$q$-binomial coefficients $\left[\begin{array}[]{c}n\\\
k\end{array}\right]=\frac{[n]!}{[n-k]![k]!}$, where $[k]=\frac{1-q^{k}}{1-q}$
and $[k]!=[1][2]\dots[k]$. The $q$-numbers appear because $\mathbf{B}$ and
$\mathbf{y}$ may not commute but they $q$-commute in the sense that
$\mathbf{B}\mathbf{y}=q\mathbf{y}\mathbf{B}$. Using the $q$-binomial theorem
in the form
$(\mathbf{y}+\mathbf{B})^{n}=\sum_{k=0}^{n}\left[\begin{array}[]{c}n\\\
k\end{array}\right]\mathbf{y}^{k}\mathbf{B}^{n-k}$ we find

$\mathbf{\Delta}(e^{\eta\mathbf{y}})=\sum_{n=0}^{\infty}\frac{\eta^{n}}{n!}(\mathbf{y}_{2}+\mathbf{y}_{1}\mathbf{B}_{2})^{n}=\sum_{n=0}^{\infty}\frac{\eta^{n}}{n!}\sum_{k=0}^{n}\left[\begin{array}[]{c}n\\\
k\end{array}\right]\mathbf{y}_{2}^{k}\mathbf{B}_{2}^{n-k}\mathbf{y}_{1}^{n-k}$

Except for the special case $\epsilon=0$ we do not know how to sum this series
to make a Gaussian expression. Instead we will expand in $\epsilon$ around
$\epsilon=0$. That way $\mathcal{G}(\Delta^{i}_{jk})$ will be a perturbed two-
step Gaussian where now the perturbation is a series in $\epsilon$. For
example to the first order in $\epsilon$ we find

$\mathcal{G}(\Delta^{i}_{jk})=e^{\beta_{i}(b_{j}+b_{k})+\eta_{i}(B_{k}y_{j}+y_{k})}(1+\frac{1}{2}B_{k}\eta_{i}^{2}y_{j}y_{k}\epsilon+\mathcal{O}(\epsilon^{2}))$

In Section 6 we will show how to compute higher order terms in $\epsilon$, for
now we will end this section by listing a similar perturbed two-step Gaussian
expression for the generating function of the antipode:

$\mathcal{G}(S_{i})=e^{-\beta_{i}b_{i}-B_{i}^{-1}\eta_{i}y_{i}}(1-(B_{i}^{-1}\beta_{i}\eta_{i}y_{i}+\frac{1}{2}B_{i}^{-2}\eta_{i}^{2}y_{i}^{2})\epsilon+\mathcal{O}(\epsilon^{2}))$

The above discussion of the co-product illustrates an important point that we
will return to often. Instead of computing with $q$-special functions such as
$q$-factorials, $q$-binomials and $q$-exponentials and so on we instead expand
$q=e^{\epsilon}=1+\epsilon+\dots$ as series in $\epsilon$. For simple formulas
expanding $q$ like this seems wasteful in that we sacrifice nice closed form
expressions for infinite series in $\epsilon$. However as the expressions one
computes with get more complicated the decisive advantage of expanding in
$\epsilon$ is that this way all the formulas are in a well understood space of
perturbed two-step Gaussians that we will formalize in Section 6.

###  5.2 Generalities on quasi-triangular Hopf algebras

Before getting to our main example we give some details on quasi-triangularity
and the Drinfeld double construction. This material is standard [5] but for
convenience we write it in our notation. Recall the multiplication
$\mathbf{m}^{ij}_{k}$ acts on tensor factors $i$ and $j$ and placing the
result in (previously unused) factor $k$. In the same way we denote by
$\mathbf{\Delta}^{i}_{jk}$ the co-product applied to factor $i$, placing the
result in factors $j$ and $k$.

######  Definition 19.

(quasi-triangular Hopf algebra)
A Hopf algebra $H$ is quasi-triangular if it contains an invertible element
$\mathbf{R}\in H^{\otimes 2}$ that satisfies the following axioms:

  1. 1.

$\mathbf{R}_{13}//\mathbf{\Delta}^{1}_{12}=\mathbf{R}_{13}\mathbf{R}_{24}//\mathbf{m}^{34}_{3}$

  2. 2.

$\mathbf{R}_{13}//\mathbf{\Delta}^{3}_{23}=\mathbf{R}_{13}\mathbf{R}_{42}//\mathbf{m}^{14}_{1}$

  3. 3.

$\mathbf{\Delta}^{i}_{kj}\mathbf{R}_{12}//\mathbf{m}^{j1}_{1}\mathbf{m}^{k2}_{2}=\mathbf{\Delta}^{i}_{jk}\mathbf{R}_{12}//\mathbf{m}^{1j}_{1}\mathbf{m}^{2k}_{2}$

As hinted at above an important consequence of quasi-triangularity is that
$\mathbf{R}$ solves the Yang-Baxter equation:

$\mathbf{F}//\mathbf{m}^{aj}_{1}//\mathbf{m}^{bi}_{2}=\mathbf{F}//\mathbf{m}^{ia}_{1}//\mathbf{m}^{jb}_{2},\qquad\mathbf{F}=\mathbf{R}_{ab}(\mathbf{R}_{1i}\mathbf{R}_{2j}//\mathbf{m}^{12}_{0})$

Another useful consequence is that the antipode inverts the $R$-matrix and
applying it on both sides has no effect:

$\mathbf{R}_{ij}^{-1}=\mathbf{R}_{ij}//\mathbf{S}_{i},\qquad\mathbf{R}_{ij}//\mathbf{S}_{i}//\mathbf{S}_{j}=\mathbf{R}_{ij}$

Applying the antipode to the ’wrong’ side of the $R$-matrix gives rise to the
Drinfeld element $\mathbf{u}$ defined by

$\mathbf{u}_{i}=\mathbf{R}_{12}//\mathbf{S}_{2}//\mathbf{m}^{21}_{i}$

######  Definition 20.

(ribbon Hopf algebra)
A ribbon element in a quasi-triangular Hopf algebra is a central element
$\mathbf{v}$ satisfying:

$\mathbf{v}^{2}=\mathbf{u}\mathbf{S}(\mathbf{u})\qquad\mathbf{\Delta}^{i}_{r,l}(\mathbf{v}_{i})=\mathbf{v}_{5}\mathbf{v}_{6}\mathbf{R}_{21}^{-1}\mathbf{R}_{43}^{-1}//\mathbf{m}^{145}_{l}//\mathbf{m}^{236}_{r}\qquad\mathbf{S}(\mathbf{v})=\mathbf{v}\qquad\boldsymbol{\varepsilon}(\mathbf{v})=1$

where $\tau(a\otimes b)=b\otimes a$. A quasi-triangular Hopf algebra together
with a choice of a ribbon element is called a ribbon Hopf algebra.

Ribbon elements do not always exist and if they do then they may not be
unique. Fortunately in the case we are interested in there does exist a
canonical choice of a ribbon element.

######  Lemma 21.

(Spinner)
A spinner in a quasi-triangular Hopf algebra $H$ is an element $\mathbf{C}\in
H$ satisfying:

$\mathbf{C}^{-1}\mathbf{u}=\mathbf{S}(\mathbf{u})\mathbf{C},\quad\boldsymbol{\varepsilon}(\mathbf{C})=1,\quad\mathbf{\Delta}(\mathbf{C})=\mathbf{C}\otimes\mathbf{C},\quad\mathbf{S}(\mathbf{C})=\mathbf{C}^{-1},\quad\mathbf{C}x\mathbf{C}^{-1}=\mathbf{S}^{2}(x)$

A ribbon Hopf algebra always comes with a choice of spinner given by
$\mathbf{C}=\mathbf{u}\mathbf{v}^{-1}$. Conversely the existence of a spinner
in $H$ implies $H$ is ribbon with ribbon element
$\mathbf{v}=\mathbf{C}^{-1}\mathbf{u}$.

In the literature the spinner $\mathbf{C}$ is known as the distinguished
group-like element but for reasons becoming clear in Section 7 prefer the name
spinner. In that section we also learn how to use the universal invariant as a
graphical calculus to manipulate the quasi-triangular Hopf algebra axioms in
terms of tangle diagrams together with a sense of planar rotation, a spinner.
In this context all of the formulas given here make intuitive sense.

###  5.3 Generalities on the Drinfeld double

In this section we briefly review the Drinfeld double construction. It is a
procedure for turning any Hopf algebra $H$ into a quasi-triangular Hopf
algebra $D(H)$. For the sake of argument we will assume throughout this
subsection that $H$ is finite dimensional over $\mathbb{Q}$. Analogues of this
construction also work for infinite dimensional topological algebras such as
$\mathbb{B}$ from Section 5.1, see [5] Chapter 12.2, but here we will not go
into this technical aspect in full generality.

Our form of the double construction will involve $H^{cop}$ the co-opposite of
Hopf algebra $H$ which is just $H$ with $\Delta$ replaced by
$\mathbf{\Delta}^{op}=\mathbf{\Delta}//\tau$ where $\tau(a\otimes b)=b\otimes
a$ and $S$ replaced by $S^{-1}$.

######  Theorem 22.

(Drinfeld double construction)
For any Hopf algebra $H$ there exists a unique Hopf algebra structure $D(H)$
on the vector space $H\otimes H^{*}$ such that $H\otimes 1$ and
$1\otimes(H^{*})^{cop}$ are sub-Hopf algebras and
$\mathbf{R}=\mathrm{id}_{H}\in H\otimes H^{*}$ makes $D(H)$ quasi-triangular.

To write down the Hopf algebra structure of $D(H)$ in terms of that of $H$
explicitly we pause to introduce some additional notation. First define
$\boldsymbol{\pi}:H^{*}\otimes H\to\mathbb{Q}$ to be the evaluation pairing,
so $\boldsymbol{\pi}(f\otimes x)=f(x)$. In working with tensor products we
will also use the notation $\boldsymbol{\pi}^{ij}$ to mean we are pairing the
$i$-th with the $j$-th tensor factors.

Since $D(H)=H\otimes H^{*}$ is a tensor product itself so we refine our index
notation for tensor factors in $D(H)^{\otimes J}$ a little. If $j\in J$ then
we denote by $j_{1}$ and $j_{2}$ the $H$ and the $H^{*}$ parts of the tensor
factor indexed by $j$. For example $h_{1_{1}}k_{2_{1}}f_{1_{2}}=h\otimes
f\otimes k\otimes 1\in D(H)\otimes D(H)$. Also denote by
$\mathbf{\Delta}^{(2)}=\mathbf{\Delta}//(\mathrm{id}\otimes\mathbf{\Delta})$.

######  Lemma 23.

(Explicit Hopf algebra structure in the double)
As a co-algebra $D(H)=H\otimes(H^{*})^{cop}$ meaning
$\mathbf{\Delta}_{D(H)}=\mathbf{\Delta}_{H}\otimes\mathbf{\Delta}^{op}_{H^{*}}$.
The antipode is given by $\mathbf{S}_{D(H)}(r\otimes
s)=\mathbf{S}^{-1}_{H^{*}}(s)\mathbf{S}_{H}(r)$ and the co-unit is
$\boldsymbol{\varepsilon}_{D(H)}=\boldsymbol{\varepsilon}_{H}\otimes\boldsymbol{\varepsilon}_{H^{*}}$.

Finally the product is given by

$(\mathbf{m}_{D(H)})^{ij}_{k}=(\mathbf{\Delta}_{H}^{(2)})^{i_{2}}_{\bar{1}\bar{2}\bar{3}}(\mathbf{\Delta}_{H^{*}}^{(2)})^{j_{1}}_{123}//(\mathbf{S}_{H^{*}}^{-1})_{3}//\boldsymbol{\pi}^{3,\bar{1}}//\boldsymbol{\pi}^{1,\bar{3}}//(\mathbf{m}_{H})^{i_{1},\bar{2}}_{k_{1}}//(\mathbf{m}_{H^{*}})^{2,j_{2}}_{k_{2}}$
(14)

###  5.4 The construction of the ribbon Hopf algebra $\mathbb{D}$

In this section we explore an appropriate dual to $\mathbb{B}$ and combine it
with $\mathbb{B}$ to construct our main example $\mathbb{D}$. In our context
of topological $\mathbb{K}$-modules the dual of $\mathbb{K}$-module $M$ is
$\mathrm{Hom}(M,\mathbb{K})$ where $\mathrm{Hom}$ means $\mathbb{K}$-module
maps.

######  Definition 24.

(The dual algebra $\mathbb{A}$)
Define $\tilde{\mathbb{B}}\subset\mathbb{B}$ to be the sub algebra
topologically generated by $\tilde{\mathbf{y}}=\hbar\mathbf{y}$ and
$\tilde{\mathbf{b}}=\hbar\mathbf{b}$. Also define
$\mathbb{A}=\tilde{\mathbb{B}}^{*}$ with the strong topology coming from the
norm $|f|=\sup_{z\neq 0}\frac{|f(z)|}{|z|}$.

On $\mathbb{A}$ there is a multiplication $\mathbf{m}_{\mathbb{A}}$ dual to
the coproduct $\Delta_{\mathbb{B}}$. In terms of the evaluation pairing

$\boldsymbol{\pi}:\mathbb{A}\otimes\mathbb{B}\to\mathbb{K}$

this means $\boldsymbol{\pi}(uu^{\prime},v)=\boldsymbol{\pi}(u\otimes
u^{\prime},\Delta(v))$ for all $u,u^{\prime}\in\mathbb{A}$ and all
$v\in\mathbb{B}$. With respect to this multiplication we would like to find
topological algebra generators for $\mathbb{A}$. Elements of
$\mathbb{A}=\mathrm{Hom}(\mathbb{B},\mathbb{K})$ are continuous so they are
uniquely defined by their values on the anti-alphabetically ordered monomials
$\mathbf{y}^{m}\mathbf{b}^{n}$. Introduce $\mathbf{a},\mathbf{x}\in\mathbb{A}$
by
$\boldsymbol{\pi}(\mathbf{a},\mathbf{y}^{m}\mathbf{b}^{n})=\delta_{m,0}\delta_{n,1}$,
where $\delta$ is the Kronecker delta. So $\mathbf{a}$ vanishes of on all
anti-alphabetically ordered monomials except for $\mathbf{b}$. Likewise define
$\mathbf{x}$ by
$\boldsymbol{\pi}(\mathbf{x},\mathbf{y}^{m}\mathbf{b}^{n})=\delta_{m,1}\delta_{n,0}$.

######  Lemma 25.

(The dual algebra $\mathbb{A}$)
If $\boldsymbol{\pi}:\mathbb{A}\otimes\tilde{\mathbb{B}}\to\mathbb{K}$ is the
evaluation pairing, introduce $\mathbf{a},\mathbf{x}\in\mathbb{A}$ by
$\boldsymbol{\pi}(\mathbf{a},\tilde{\mathbf{y}}^{m}\tilde{\mathbf{b}}^{n})=\delta_{m,0}\delta_{n,1}$
and
$\boldsymbol{\pi}(\mathbf{x},\tilde{\mathbf{y}}^{m}\tilde{\mathbf{b}}^{n})=\delta_{m,1}\delta_{n,0}$.
$\mathbb{A}$ is topologically generated over $\mathbb{K}$ by
$\mathbf{a},\mathbf{x}$. The evaluation pairing $\boldsymbol{\pi}$ satisfies

$\boldsymbol{\pi}(\mathbf{a}^{k}\mathbf{x}^{\ell},\tilde{\mathbf{y}}^{m}\tilde{\mathbf{b}}^{n})=\delta_{k,n}\delta_{m,n}[m]!n!$

Moreover the generators satisfy the commutation relation

$[\mathbf{a},\mathbf{x}]=\mathbf{x}$

$\mathbb{A}$ is of PBW-type using
$\mathbb{O}:\mathbb{Q}_{\hbar}[\epsilon,a,x]\to\mathbb{A}$ sending monomials
to alphabetically ordered monomials.

###### Proof.

(Sketch, see [4] section 8.3B for the full account).
As an illustration of the general argument, let us find out what linear map is
represented by $\mathbf{x}^{2}$. We have to pair it with all ordered monomials
$\mathbf{y}^{n}\mathbf{b}^{m}$. By definition

$\boldsymbol{\pi}(\mathbf{x}^{2},\mathbf{y}^{n}\mathbf{b}^{m})=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{x}_{2},\Delta(\mathbf{y}^{n}\mathbf{b}^{m}))=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{x}_{2},\Delta(\mathbf{y})^{n}\Delta(\mathbf{b})^{m})$

In the previous section we learnt that
$\Delta(\mathbf{y})^{n}=\sum_{k=0}^{n}\left[\begin{array}[]{c}n\\\
k\end{array}\right]\mathbf{y}_{2}^{k}\mathbf{B}_{2}^{n-k}\mathbf{y}_{1}^{n-k}$
and similarly $\Delta(\mathbf{b})^{m}=(\mathbf{b}_{1}+\mathbf{b}_{2})^{m}$ .
In the pairing $\boldsymbol{\pi}(\mathbf{x}^{2},\mathbf{y}^{n}\mathbf{b}^{m})$
we are supposed to pair the variables with common indices and so the pairing
can only be non-zero when $m=0$ because each term in $\Delta(\mathbf{b})^{m}$
contains at least one of $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$. By definition
$\mathbf{x}_{i}$ only has a non-zero pairing with $\mathbf{y}_{i}$ so the only
contribution comes from $k=1=n-k$ giving
$\boldsymbol{\pi}(\mathbf{x}^{2},\mathbf{y}^{n}\mathbf{b}^{m})=[2]$.

The appearance of the $q$-factorial can be understood by viewing $[n]!$ as the
generating function of the numbers permutations with a fixed number of
inversions. By an inversion of $\sigma$ we mean a pair $i<j$ such that
$\sigma(i)>\sigma(j)$ and denote the number of inversions of $\sigma$ by
$\mathrm{inv}(\sigma)$. The result is $[n]!=\sum_{\sigma\in
S_{n}}q^{\mathrm{inv}(\sigma)}$.

In
$\boldsymbol{\pi}(\mathbf{x}^{n},\mathbf{y}^{n})=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{x}_{2}\dots\mathbf{x}_{n},\Delta^{(n-1)}(\mathbf{y})^{n})$
this will come up because (writing $\Delta^{(s)}$ for the repeated application
of the coproduct):

$\Delta^{(n-1)}(\mathbf{y})^{n}=\big{(}\sum_{k=1}\mathbf{y}_{k}\prod_{j=1+k}^{n}\mathbf{B}_{j}\big{)}^{n}$

Pairing this with $\mathbf{x}_{1}\mathbf{x}_{2}\dots\mathbf{x}_{n}$ we get
precisely one term for each permutation $\sigma\in S_{n}$ because we need to
select each of the terms in $\Delta^{(n-1)}(\mathbf{y})$ precisely once. Each
term needs to be brought into anti-alphabetic order before pairing and the
number of commutations necessary is precisely $q^{\mathrm{inv}(\sigma)}$.

Similarly one checks the commutation relation between $\mathbf{a},\mathbf{x}$
by computing the pairing of $\mathbf{x}\mathbf{a}$ with all monomials
$\mathbf{y}^{n}\mathbf{b}^{m}$ to find that the only non-zero pairings are
$\boldsymbol{\pi}(\mathbf{x}\mathbf{a},\mathbf{y}\mathbf{b})=1$ and

$\boldsymbol{\pi}(\mathbf{x}\mathbf{a},\mathbf{y})=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{a}_{2},\Delta(\mathbf{y}))=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{a}_{2},\mathbf{y}_{2}+\mathbf{y}_{1}\mathbf{B}_{2}))=\boldsymbol{\pi}(\mathbf{x}_{1}\mathbf{a}_{2},-\mathbf{y}_{1}\mathbf{b}_{2}))=-1$

∎

######  Lemma 26.

(The Hopf algebra $\mathbb{A}$)
The following determines a Hopf algebra structure on $\mathbb{A}$ that makes
$\boldsymbol{\pi}$ a pairing of Hopf algebras.
$\varepsilon(\mathbf{a})=\varepsilon(\mathbf{x})=0$ and
$[\mathbf{a},\mathbf{x}]=\mathbf{x}$ and setting
$\mathbf{A}=e^{-\epsilon\hbar\mathbf{a}}$ we have

$\mathbf{\Delta}(\mathbf{a})=\mathbf{a}_{1}+\mathbf{a}_{2}\quad\mathbf{S}(\mathbf{a})=-\mathbf{a}\quad\mathbf{\Delta}(\mathbf{x})=\mathbf{x}_{1}+\mathbf{A}_{1}\mathbf{x}_{2}\quad\mathbf{S}(\mathbf{x})=-\mathbf{A}^{-1}\mathbf{x}$

###### Proof.

All these relations can be established by patiently pairing with the monomial
basis, see [4] section 8.3B for details. ∎

With the Drinfeld double construction in mind we define our main example
$\mathbb{D}$:

######  Definition 27.

(The $\mathbb{K}$-module $\mathbb{D}$)
As $\mathbb{K}$-modules define $\mathbb{D}=\mathbb{B}\otimes\mathbb{A}$ and
$\tilde{\mathbb{D}}=\tilde{\mathbb{B}}\otimes\mathbb{A}$. On
$\tilde{\mathbb{D}}$ define a Hopf algebra structure using the formulas for
$D(\tilde{\mathbb{B}})$ from Subsection 5.3.

######  Lemma 28.

(The Hopf algebra $\mathbb{D}$)
The Hopf algebra structure of $\tilde{\mathbb{D}}$ extends to $\mathbb{D}$
making it $\hbar$-adically generated by
$\mathbf{y},\mathbf{b},\mathbf{a},\mathbf{x}$ with relations

$\mathbf{x}\mathbf{y}=q\mathbf{y}\mathbf{x}+\frac{\mathbf{1}-\mathbf{AB}}{\hbar}\quad[\mathbf{a},\mathbf{x}]=\mathbf{x}\quad[\mathbf{b},\mathbf{x}]=\epsilon\mathbf{x}\quad[\mathbf{a},\mathbf{y}]=-\mathbf{y}\quad[\mathbf{b},\mathbf{y}]=-\epsilon\mathbf{y}\quad[\mathbf{a},\mathbf{b}]=0$

The coproduct is determined by
$\mathbf{\Delta}_{\mathbb{D}}=\mathbf{\Delta}_{\mathbb{B}}\otimes\mathbf{\Delta}^{op}_{\mathbb{A}}$,
where $\mathbf{\Delta}^{op}=\mathbf{\Delta}//\tau$ and $\tau(r\otimes
s)=s\otimes r$. The antipode is given by $\mathbf{S}_{\mathbb{D}}(r\otimes
s)=\mathbf{S}^{-1}_{\mathbb{A}}(s)\mathbf{S}_{\mathbb{B}}(r)$ and the co-unit
$\varepsilon_{\mathbb{D}}$ just sends all four generators to $0$.

###### Proof.

(sketch, see [4] Section 8.3B for details).

The commutation relations between the generators can be derived from formula
(14). The most interesting one is

$\mathbf{x}\tilde{\mathbf{y}}=\mathbf{x}_{i}\tilde{\mathbf{y}}_{j}//(\mathbf{m}_{\tilde{\mathbb{D}}})^{ij}_{k}=q\tilde{\mathbf{y}}\mathbf{x}+1-\mathbf{AB}$

To derive this we first compute

$\displaystyle\mathbf{\Delta}_{\mathbb{A}}^{(2)}(\mathbf{x})//(\mathbf{S}^{-1}_{\mathbb{A}})_{3}$
$\displaystyle=\mathbf{x}_{1}+\mathbf{A}_{1}\mathbf{x}_{2}-q^{-1}\mathbf{A}_{1}\mathbf{A}_{2}\mathbf{A}_{3}^{-1}\mathbf{x}_{3}$
(15) $\displaystyle\mathbf{\Delta}^{(2)}(\tilde{\mathbf{y}})$
$\displaystyle=\tilde{\mathbf{y}}_{3}+\mathbf{B}_{3}\tilde{\mathbf{y}}_{2}+\mathbf{B}_{3}\mathbf{B}_{2}\tilde{\mathbf{y}}_{1}$
(16)

The formula says that we need to pair the 1st and 3rd factor of the top three
monomials with the 3rd and the 1st tensor factor of the bottom three
monomials. When we get a non-zero answer we should multiply it with whatever
is present in the 2nd tensor factors. Since $\mathbf{x}$ only pairs non-zero
with $\tilde{\mathbf{y}}$ each monomial can only get a non-zero pairing with
the one right below it in Equation (15). Using $\pi(\mathbf{A},\mathbf{B})=q$
the three terms contribute $q\tilde{\mathbf{y}}\mathbf{x}$, $1$ and
$-\mathbf{AB}$ respectively. Adding these up gives the promised commutation
relation in $\tilde{\mathbb{D}}$. Dividing through by $\hbar$ we get the
corresponding relation in $\mathbb{D}$ because $1-\mathbf{AB}$ is divisible by
$\hbar$. In a similar way we find the remaining commutation relations and so
the algebra structure of $\mathbb{D}$. ∎

The main point of the Drinfeld double construction is to produce quasi-
triangular Hopf algebras. We will show below that the above defined relations
on $\mathbb{D}$ are in fact obtained by the double construction. Hence we gain
an $R$-matrix and in this case even a ribbon element.

######  Theorem 29.

($\mathbb{D}$ is a ribbon Hopf algebra)
$\mathbb{D}$ is a ribbon Hopf algebra with respect to $\mathbf{R},\mathbf{C}$
defined below

$\mathbf{R}=\sum_{m,n=0}^{\infty}\frac{\hbar^{m+n}}{[m]!n!}\mathbf{y}^{m}\mathbf{b}^{n}\otimes\mathbf{a}^{n}\mathbf{x}^{m}$

and

$\mathbf{C}=(\mathbf{AB})^{\frac{1}{2}}$

where we take the positive square root in the sense of power series in
$\hbar$.

###### Proof.

The first part follows from the topological version of the double
construction, see [5] Section 12.2 and [4] Section 8.3B. We verified above
that the algebra structure for $\mathbb{D}$ comes from the double
construction. That way the formula for the universal $R$-matrix is automatic
as we already decided $\tilde{\mathbf{y}}^{m}\tilde{\mathbf{b}}^{n}$ is a
basis for $\tilde{\mathbb{B}}$ with dual basis
$\frac{1}{[m]!n!}\mathbf{a}^{n}\mathbf{x}^{m}$ for $\mathbb{A}$. Finally the
formula for $\mathbf{C}$ follows from the fact that
$\mathbf{u}=\mathbf{A}\mathbf{B}\ \mathbf{S}(\mathbf{u})$ which is proven in
Appendix A. ∎

We end this section with a comment on the relation to one of the most common
quantum groups: $\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$. Recall this is a
quantization of the universal enveloping algebra of $\mathfrak{sl}_{2}$, [4].
It has (topological) generators $H,E,F$ over $\mathbb{Q}[[\hbar]]$ with
relations

$[H,E]=2E\quad[H,F]=-2F\quad[E,F]=\frac{q^{\frac{H}{2}}-q^{-\frac{H}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}\qquad
q=e^{\hbar}$

In fact we have the following isomorphism:

######  Lemma 30.

Set $\epsilon=1$ and define
$\phi:\mathbb{D}\to\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$ by
$\phi(\mathbf{a})=\frac{1}{2}H$ and
$\phi(\mathbf{x})=\frac{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}{q\hbar}Eq^{-H/2}$
and $\phi(\mathbf{y})=F$ and $\phi(\mathbf{t})=0$ provides a ribbon Hopf
algebra isomorphism between $\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$ and the
quotient of $\mathbb{D}$ by the two sided ideal generated by the central
element $\mathbf{t}=\mathbf{b}-\epsilon\mathbf{a}$.

###### Proof.

Instead of proving the isomorphism formally we remark that the construction of
$\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$ as carried out in e.g. Chapter 8 of
[4] takes precisely the same steps we took to construct $\mathbb{D}$. The only
difference is that to obtain $\mathcal{U}_{\hbar}(\mathfrak{sl}_{2})$ one
assumes $\epsilon=1$ and, quotients by $\mathbf{t}$ and makes a slight change
of variables at the very end. The lemma is thus true by construction. ∎

##  6 Gaussian generating functions for $\mathbb{D}$

In this section we apply the generating function ideas developed in Section 2
to our main example $\mathbb{D}$. Ordering the generators $\mathbf{y,b,a,x}$
we get an isomorphism of topological $\mathbb{K}$-modules
$\mathbb{O}:\mathbb{Q}_{\hbar}[\epsilon,y,b,a,x]\to\mathbb{D}$. More precisely
$\mathbb{O}$ is defined on monomials as follows and is extended
$\mathbb{K}$-linearly.

$\mathbb{O}(y^{k}b^{\ell}a^{m}x^{n})=\mathbf{y}^{k}\mathbf{b}^{\ell}\mathbf{a}^{m}\mathbf{x}^{n}$

In the terminology of Definition 16 the algebra $\mathbb{D}$ is of PBW type.
We use the same notation also for the extension of this map to tensor products
indexed by a finite set $J$ to get an isomorphism

$\mathbb{O}:\mathbb{Q}_{\hbar}[\epsilon,y_{J},b_{J},a_{J},x_{J}]\to\mathbb{D}^{\otimes
J}$

Extending the categories introduced in Section 2 to $\mathbb{K}$ we get
categories $\mathcal{D},\mathcal{H},\mathcal{C}$ to describe
$\mathbb{K}$-linear maps between tensor powers of $\mathbb{D}$. These
categories all have the same objects, i.e. finite sets and their morphisms are
defined as follows:

  1. 1.

$\mathrm{Hom}_{\mathcal{D}}(J,K)=\mathrm{Hom}(\mathbb{D}^{\otimes
J},\mathbb{D}^{\otimes K})$,

  2. 2.

$\mathrm{Hom}_{\mathcal{H}}(J,K)=\mathrm{Hom}(\mathbb{Q}_{\hbar}[\epsilon,y_{J},b_{J},a_{J},x_{J}],\mathbb{Q}_{\hbar}[\epsilon,y_{K},b_{K},a_{K},x_{K}])$,

  3. 3.

$\mathrm{Hom}_{\mathcal{C}}(J,K)=\mathbb{Q}_{\hbar}[\epsilon,y_{K},b_{K},a_{K},x_{K}][[\eta_{J},\beta_{J},\alpha_{J},\xi_{J}]]$.

We then have two functors both of which are equivalences of categories. The
first functor $\mathcal{O}:\mathcal{D}\to\mathcal{H}$ expresses the given map
in terms of the basis and is the identity on objects. So it maps
$\mathbf{f}:\mathbb{D}^{\otimes J}\to\mathbb{D}^{\otimes K}$ to $f$ where
$f(z)=(\mathbb{O}^{\otimes K})^{-1}(\mathbf{f}(\mathbb{O}^{\otimes J}(z)))$.
The second functor $\mathcal{G}:\mathcal{H}\to\mathcal{C}$ is takes the
generating function as in Section 2. It is the exponential generating function
of all values of the morphism $f\in\mathrm{Hom}_{\mathcal{H}}(J,K)$ on the
monomials. More precisely

$\mathcal{G}(f)=f(e^{\eta_{J}y_{K}+\tau_{J}b_{K}+\alpha_{J}a_{K}+\xi_{J}x_{K}})$

Composition in the categories $\mathcal{H},\mathcal{C}$ is designed so as to
make sure $\mathcal{O},\mathcal{G}$ are really functors. So by definition
$f//g=\mathbb{O}^{-1}(\mathbb{O}(f)//\mathbb{O}(g))$ describes composition in
$\mathcal{H}$. More importantly composition in $\mathcal{C}$ comes out as

$\psi//\phi=\psi\phi|_{\zeta_{K}\mapsto\partial_{z_{K}}}|_{z_{K}=0}=\langle\psi\phi\rangle_{z_{K}}$

Here $z$ is short-hand for all generators $z_{K}=(y_{K},t_{K},a_{K},x_{K})$
and $\zeta_{K}=(\eta_{K},\tau_{K},\alpha_{K},\xi_{K})$.

The generating functions for the tangle invariants and Hopf operations in
$\mathbb{D}$ that we will be interested in turn out to belong to a very
special and small subcategory of $\mathcal{C}$. This category is the category
$\mathcal{PG}$ of perturbed Gaussian generating functions. The definition of
$\mathcal{PG}$ may at first seem rather technical but the fact that its
morphisms are given by concrete formulas makes them much more tractable in
practice. When one tries to do similar computations in a regular quantum group
the $q$-hypergeometric expressions often become hard to manage rather quickly.
Unlike in $\mathcal{PG}$ there seems to be no notion of closure, no useful
specific form that all expressions must take in usual the quantum group
setting.

######  Definition 31.

(Perturbed Gaussians $\mathcal{PG}$)
Define two weights $\mathrm{wt},\mathrm{wh}$ on the monomials in $\mathcal{C}$
by $\mathrm{wt}(uv)=\mathrm{wt}(u)+\mathrm{wt}(v)$ and
$\mathrm{wh}(uv)=\mathrm{wh}(u)+\mathrm{wh}(v)$ and the following values on
the generators:

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c}{}\hfil&y&\eta&b&\beta&a&\alpha&x&\xi&\epsilon&\hbar\\\
\hline\cr\mathrm{wh}&1&-1&1&-1&0&0&0&0&1&-1\\\
\mathrm{wt}&1&1&0&2&2&0&1&1&-4&0\end{array}$

Also define $\mathcal{A}=e^{\alpha}$. For finite sets $J,K$ define
$\mathcal{PG}(J,K)$ to be the elements $Pe^{G}\in\mathcal{C}(J,K)$ satisfying:

  1. 1.

$\mathrm{wh}$ is zero on all monomials in $P$ and $G$.

  2. 2.

$\mathrm{wt}(G)=2$ and for all monomials in $P$ we have $\mathrm{wt}\leq 0$.

  3. 3.

$G=(\beta_{J},a_{K})G^{(1)}{b_{K}\choose\alpha_{J}}+(y_{K},\xi_{J})G^{(2)}{\eta_{J}\choose
x_{K}}$ with $G^{(1)}_{jk}\in\mathbb{Q}[\hbar]$ and
$G^{(2)}_{jk}\in\mathbb{Q}(\mathcal{A}_{J},B_{K}^{\frac{1}{2}})[\hbar]$.

  4. 4.

$P=\sum_{k=0}^{\infty}P_{k}\epsilon^{k}$ with
$P_{k}\in\mathbb{Q}(\mathcal{A}_{J},B_{K}^{\frac{1}{2}})[z_{K},\zeta_{J},\hbar]$.

Here we abbreviated $z=(y,b,a,x)$ and $\zeta=(\eta,\beta,\alpha,\xi)$.

The formula for $G$ above should be read as matrix and vector multiplication,
e.g. $(\beta_{J},a_{K})$ represents a row vector and $G^{(1)}=(G^{(1)}_{jk})$
is a matrix of size $|J|+|K|$. More precisely we have

$(\beta_{J},a_{K})G^{(1)}{b_{K}\choose\alpha_{J}}=\sum_{i,j\in
J}\beta_{i}G^{(1)}_{i,j}\alpha_{j}+\sum_{j\in J,k\in
K}\beta_{j}G^{(1)}_{j,k}b_{k}+\sum_{j\in J,k\in
K}a_{k}G^{(1)}_{k,j}\alpha_{j}+\sum_{j\in J,k\in
K}a_{k}G^{(1)}_{k,\ell}b_{\ell}$

The above definition is motivated by looking closely at the expansion of the
R-matrix and the multiplication in $\mathbb{D}$. Parts of it can already be
observed in the two-step Gaussians from Section 5.1. Notice that the
$\hbar$-dependence of $P$ and $G$ is completely determined by the condition
$\mathrm{wh}=0$. For example the Gaussian exponent $G^{(1)}$ is at most linear
in $\hbar$.

For technical reasons we introduce two more flavours of the $\mathcal{PG}$.
These will be important when dealing with the double construction and
controlling the denominators of the knot invariant in later sections.

######  Definition 32.

(Perturbed Gaussians $\mathcal{PG}^{\pm},\mathcal{PG}^{+}$)
For finite sets $J,K$ define
$\mathcal{C}^{\pm}(J,K)=\mathbb{Q}[\epsilon,z_{K},\hbar^{\pm 1}][[\zeta_{J}]]$
and extend $\mathrm{wt}$ by $\mathrm{wt}(\hbar^{-1})=1$. Also define
$\mathcal{PG}^{\pm}(J,K)$ to be the set of elements
$Pe^{G}\in\mathcal{C}^{\pm}(J,K)$ satisfying points 1,2,3,4 of Definition 31
with the modifications that $G^{(2)}_{jk}\in\mathbb{Q}[\mathcal{A}^{\pm
1}_{J},B_{K}^{\pm\frac{1}{2}},\hbar^{\pm 1}]$ and
$P_{k}\in\mathbb{Q}[\mathcal{A}^{\pm
1}_{J},B_{K}^{\pm\frac{1}{2}},z_{K},\zeta_{J},\hbar^{\pm 1}]$.
Finally $\mathcal{PG}^{+}(J,K)$ is the set of elements of $\mathcal{PG}^{\pm}$
that do not have negative powers of $\hbar$.

For example
$e^{\alpha_{1}\beta_{2}\hbar^{-1}}\in\mathcal{PG}^{\pm}(\\{1,2\\},\emptyset)$
but not in $\mathcal{PG}^{+}$. Also note that $e^{b_{i}a_{j}\hbar}$ is not in
$\mathcal{PG}^{\pm}$ since it is not a power series in the Greek variables, it
is not in $\mathcal{C}^{\pm}$. On the other hand we do know that
$\mathcal{PG}^{+}(J,K)\subset\mathcal{PG}(J,K)$.

The reader is warned that it is not always possible to compose a morphism of
$\mathcal{PG}$ with a morphism from $\mathcal{PG}^{\pm}$. For example
$e^{a_{i}b_{j}\hbar}//e^{\alpha_{i}\beta_{j}\hbar^{-1}}$ would be infinite.
The contraction lemma does show that $\mathcal{PG}^{\pm},\mathcal{PG}^{+}$ and
$\mathcal{PG}$ are categories:

######  Lemma 33.

(Closure under composition in the $\mathcal{PG}$s)

  1. 1.

Taking the same composition law for $\mathcal{PG}^{\pm},\mathcal{PG}^{+}$ as
the one in $\mathcal{PG}$, all three are closed under composition.

  2. 2.

Suppose $f\in\mathcal{PG}(J,K)$ and $g\in\mathcal{PG}^{\pm}(K,L)$. If
$f|_{\epsilon=0}//g|_{\epsilon=0}\in\mathcal{PG}^{\pm}(J,L)$ is well-defined
then $f//g\in\mathcal{PG}^{\pm}(J,L)$ too.

###### Proof.

We will show that for $I,J,K$ disjoint and $\phi\in\mathcal{PG}(I,J)$ and
$\psi\in\mathcal{PG}(J,K)$ we have
$\phi//\psi=\langle\phi\psi\rangle_{z_{J},\zeta_{J}}\in\mathcal{PG}(I,K)$. By
definition $\phi$ and $\psi$ are morphisms in the category $\mathcal{C}$ and
so their composition is a morphism of $\mathcal{C}(I,K)$.

Since $\phi\psi\in\mathcal{PG}(I\cup J,J\cup K)$ it suffices to show the
following simpler result. If $v\in\mathcal{PG}(I\cup\\{j\\},K\cup\\{j\\})$ is
such that there exists $u\in\mathcal{C}(I,K)$ with $u=\langle
v\rangle_{z_{j},\zeta_{j}}$ then we must have $u\in\mathcal{PG}(I,K)$. The
four contractions to get from $v$ to $u$ may be carried one at a time and for
each of the contractions it is easy to see that the conditions on $P$ and $G$
are preserved using the one variable version of the contraction theorem, see
Equation (4).

Moving on to $\mathcal{PG}^{\pm}$ the proof is the same except that we have to
argue that no denominators will arise because the $W$ in Equation (4) is
always $0$. Finally the case $\mathcal{PG}^{+}$ follows from the observation
that no inverse powers of $\hbar$ can be created using this formula.

Part 2) Is proven by again looking at the contraction formula and noting that
the only way the contraction can be undefined is for the determinant to
vanish. If that happens it will already have happened at $\epsilon=0$. ∎

With the definitions in place we now set out to prove that all structural
elements and maps of $\mathbb{D}$ are in $\mathcal{PG}$. Recall we use bold
notation for elements in $\mathbb{D}$ and regular script for the corresponding
element of $\mathcal{C}$. For example
$R_{ij}=\mathbf{R}_{ij}//\mathbb{O}^{-1}$.

######  Theorem 34.

(Main $\mathcal{PG}$ theorem)
All structural elements and maps of $\mathbb{D}$ are in $\mathcal{PG}$. More
precisely:

  1. 1.

$R_{ij}^{\pm 1}\in\mathcal{PG}(\emptyset,\\{i,j\\})$ and
$v_{i}^{\pm},C_{i}^{\pm 1}\in\mathcal{PG}(\emptyset,\\{i\\})$.

  2. 2.

The generating functions
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{D}},\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}},\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{D}},\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\varepsilon_{\mathbb{D}}$
for the Hopf algebra operations of $\mathbb{D}$ are morphisms of
$\mathcal{PG}^{+}$.

  3. 3.

$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{ij}\in\mathcal{PG}^{\pm}(\\{i,j\\},\emptyset)$.

The remainder of this section is dedicated to the proof of the main
$\mathcal{PG}$ theorem. We break it down in a lemma for each of the elements
and operations, starting with the $R$-matrix.

######  Lemma 35.

(Faddeev [19])
Recall $q=e^{\hbar\epsilon}$ and
$e_{q}^{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{[n]!}$. We have

$e_{q}^{z}=e^{z}\exp\sum_{n=2}^{\infty}\frac{(1-q)^{n}z^{n}}{(1-q^{n})n}$

and
$R_{ij}=\mathbb{O}^{-1}(\mathbf{R}_{ij})=e^{b_{i}a_{j}\hbar}e_{q}^{y_{i}x_{j}\hbar}\in\mathcal{PG}(\emptyset,\\{{i,j\\}})$.

###### Proof.

Following Zagier [19], we recall the $q$-derivative of a function $f(z)$ is
$\frac{f(qz)-f(z)}{qz-z}$. From the series formula of the $q$-exponential is
easily checked that it equals its $q$-derivative so

$e_{q}^{z}=\frac{e_{q}^{qz}-e_{q}^{z}}{qz-z}\quad\text{or rearranging}\quad
e_{q}^{qz}=e_{q}^{z}(1+(q-1)z)$

Taking the logarithm on both sides yields $\log e_{q}^{qz}=\log
e_{q}^{z}+\log(1-(1-q)z)$. If we set $\log e_{q}^{z}=\sum_{n}c_{n}z^{n}$ and
compare powers of $z^{n}$ in the previous equation then we find
$q^{n}c_{n}=c_{n}-\frac{(1-q)^{n}}{n}$. It follows that
$c_{n}=\frac{(1-q)^{n}z^{n}}{(1-q^{n})n}$ and exponentiating proves the first
assertion of the lemma.

The formula for the R-matrix from Theorem 29 implies that
$R_{ij}=\mathbb{O}^{-1}(\mathbf{R}_{ij})=e^{b_{i}a_{j}\hbar}e_{q}^{y_{i}x_{j}\hbar}$.
To show that it is in $\mathcal{PG}(\emptyset,\\{i,j\\})$ we write
$R_{ij}=Pe^{G}$ with $G=\hbar(b_{i}a_{j}+y_{i}x_{j})$. Clearly
$\mathrm{wh}(G)=0$ and $\mathrm{wt}(G)=2$. The expression for $e_{q}^{z}$
proven in the first part of the lemma shows that the perturbation part of
$R_{ij}$ is

$P=\exp\sum_{n=2}^{\infty}\frac{(1-q)^{n}(y_{i}x_{j}\hbar)^{n}}{(1-q^{n})n}=1+\sum_{k=1}^{\infty}P_{k}\epsilon^{k}$

from which it is already clear that $\mathrm{wh}(P_{k})=0$. We should also
check that $\mathrm{wt}(P_{k})\leq 0$ and that its coefficients do not depend
on $B,\mathcal{A}$ and are polynomial in the other variables. This follows by
studying the expansion of $\frac{(1-q)^{n}z^{n}}{n(1-q^{n})}$ in $\epsilon$,
recalling that $q=e^{\epsilon\hbar}$. For $n\geq 2$ this rational function has
an $(n-1)$-fold zero at $\epsilon=0$ and so the highest power of $z$ in the
coefficient of $\epsilon^{k}$ of $e_{q}^{z}$ is $2k$, coming from the taking
the exp of the $n=2$ terms. ∎

Next is the inverse of the $R$-matrix, for which we also need the generating
functions of multiplication in $\mathbb{B}$ and $\mathbb{A}$.

######  Lemma 36.

(Multiplication in $\mathbb{A},\mathbb{B}$ and $R^{-1}$)

  1. 1.

The generating functions for multiplication in $\mathbb{A}$ and $\mathbb{B}$
are in $\mathcal{PG}^{+}(\\{i,j\\},\\{k\\})$:

$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{ij}_{k}=e^{(\alpha_{i}+\alpha_{j})a_{k}+(\mathcal{A}_{j}^{-1}\xi_{i}+\xi_{j})x_{k}}\qquad(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{ij}_{k}=e^{(\beta_{i}+\beta_{j})b_{k}+(e^{-\epsilon\beta_{i}}\eta_{i}+\eta_{j})y_{k}}$

  2. 2.

There is a series $P=1+\sum_{k=1}^{\infty}P_{k}\epsilon^{k}$ such that
$R_{ij}^{-1}=Pe^{G}\in\mathcal{PG}(\emptyset,\\{i,j\\})$ with
$G=-b_{i}a_{j}-B_{i}^{-1}y_{i}x_{j}$.

###### Proof.

Part 1) already appeared as Lemma 18 in the case of $\mathbb{B}$. The proof
for the $\mathbb{A}$ case is analogous.

For part 2) we set $G=-b_{i}a_{j}-B_{i}^{-1}y_{i}x_{j}$ and find
$P=1+P_{1}\epsilon+P_{2}\epsilon^{2}+\dots$ order by order in $\epsilon$ such
that $R_{ij}^{-1}=Pe^{G}$ satisfies the defining equation

$R_{12}R_{ij}^{-1}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1i}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2j}_{j}=1_{i}1_{j}$

First we check that when $\epsilon=0$ the defining equation is true with
$R_{ij}^{-1}=e^{G}$. Note the multiplication in $\mathbb{B}$ becomes
commutative at $\epsilon=0$ so we compute (using $\langle
e^{sz_{j}}f(\zeta_{j})\rangle_{j}=f(s)$):

$R_{12}e^{G_{ij}}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1i}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2j}_{j}=\langle
e^{b_{i}a_{2}-b_{i}a_{j}+y_{i}x_{2}-B_{i}^{-1}y_{i}x_{j}+(\alpha_{2}+\alpha_{j})a_{k}+(\mathcal{A}_{j}^{-1}\xi_{2}+\xi_{j})x_{k}}\rangle_{2,j}=$
$\langle
e^{-B_{i}^{-1}y_{i}x_{j}+(B_{i}^{-1}y_{i}+\xi_{j})x_{k}}\rangle_{j}=1$

Next we compute the perturbation $P_{k}$ order by order recursively. Let us
assume that we found $P_{k}$ for all $k<n$ satisfying

$R_{12}(\sum_{k=0}^{n-1}P_{k}\epsilon^{k}e^{G})_{ij}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1i}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2j}_{j}=1_{i}1_{j}+\epsilon^{n}E_{ij}$

for some error $E_{ij}=E[0]_{ij}+\mathcal{O}(\epsilon)$. We are looking for
$P_{n}$ that satisfies

$R_{12}(Pe^{G})_{ij}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1i}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2j}_{j}=R_{12}(P_{n}\epsilon^{n}e^{G})_{IJ}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1I}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2J}_{j}+1_{i}1_{j}+\epsilon^{n}E_{ij}=0\mod\epsilon^{n+1}$

Taking the coefficient of $\epsilon^{n}$ on both sides and using
$R_{ij}=R[0]_{ij}+\mathcal{O}(\epsilon)$ yields:

$R[0]_{12}(P_{n}e^{G})_{IJ}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{1I}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2J}_{j}+E[0]_{ij}=0$
(17)

Therefore left-multiplying with $e^{G}$ yields $P_{n}e^{G}$ and multiplying by
$e^{-G}$ in the commutative sense shows $(P_{n})_{ij}=$

$e^{-G_{ij}}(e^{G})_{34}R[0]_{12}(P_{n}e^{G})_{IJ}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{31I}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{42J}_{j}=-e^{-G_{ij}}(e^{G})_{34}E[0]_{IJ}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{3I}_{i}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{4J}_{j}$

The final equality follows from associativity and solving for $E[0]$ in
Equation (17). By induction on $n$ we then see that $P_{n}$ satisfies the
criteria for $R_{ij}^{-1}$ to be in $\mathcal{PG}$. ∎

The pairing
$\boldsymbol{\pi}:\mathbb{A}\otimes\tilde{\mathbb{B}}\to\mathbb{K}$ from Lemma
25 can be extended to a $\mathbb{K}$-module map
$\boldsymbol{\pi}:\mathbb{A}\otimes\mathbb{B}\to\mathbb{Q}[\epsilon]((\hbar))$
by the formula
$\boldsymbol{\pi}(\mathbf{a}^{n}\mathbf{x}^{m},\mathbf{y}^{m^{\prime}}\mathbf{b}^{n^{\prime}})=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\frac{n![m]!}{\hbar^{m+n}}$.
Writing $\pi=\mathcal{O}(\boldsymbol{\pi})$ its generating function
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{ij}=\pi(e^{\alpha_{i}a_{i}+\xi_{i}x_{i}+\eta_{j}y_{j}+\beta_{j}y_{j}})$
can be computed order by order in $\epsilon$ and this shows it is in
$\mathcal{PG}^{\pm}$.

######  Lemma 37.

(Generating function for the pairing)
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{jk}=Pe^{G}\in\mathbb{Q}[\epsilon,\hbar^{\pm
1}][[\alpha_{j},\xi_{j},\beta_{k},\eta_{k}]]$, for
$G={\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}$ and some
$P=1+\sum_{\ell=1}^{\infty}P_{\ell}\epsilon^{\ell}$. Moreover
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{jk}\in\mathcal{PG}^{-}(\\{j,k\\},\emptyset)$.

###### Proof.

With $G$ as above we will find the coefficients $P_{\ell}$ of the perturbation
$P$ order by order such that the defining equation holds:

$R_{ij}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{jk}=e^{\hbar(\beta_{k}b_{i}+\eta_{k}y_{i})}$

At least when $\epsilon=0$ this equation is true because
$R_{ij}=e^{b_{i}a_{j}\hbar+y_{i}x_{j}\hbar}+\mathcal{O}(\epsilon)$ so

$R_{ij}//e^{\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}=e^{\hbar(\beta_{k}b_{i}+\eta_{k}y_{i})}+\mod\epsilon$

Since $\mathrm{wh}(\hbar)=1$ we see $\mathrm{wh}(G)=0$ and its coefficients
are in $\mathbb{Q}[\hbar^{\pm}]$ viewed as a quadratic in the Greek variables.

Next, assume that we computed $P_{\ell}$ for all $\ell<n$ such that for some
$E_{ik}$ independent of $\epsilon$ we have

$R_{ij}//(\sum_{\ell=0}^{n-1}P_{\ell}\epsilon^{\ell})e^{\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}=e^{\hbar(\beta_{k}b_{i}+\eta_{k}y_{i})}+\epsilon^{n}E_{ik}+\mathcal{O}(\epsilon^{n+1})$

To find $P_{n}$ we write the same equation, truncating the series for
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi$ at $\epsilon^{n}$:

$R_{ij}//(\sum_{\ell=0}^{n}P_{\ell}\epsilon^{\ell})e^{\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}=e^{\hbar(\beta_{k}b_{i}+\eta_{k}y_{i})}+R_{ij}//\big{(}P_{n}\epsilon^{n}e^{\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}+\epsilon^{n}E_{ik}\big{)}+\mathcal{O}(\epsilon^{n+1})$

Taking the coefficient of $\epsilon^{n}$ should give

$0=R[0]_{ij}//\big{(}P_{n}e^{\frac{\alpha_{j}\beta_{k}}{\hbar}+\frac{\xi_{j}\eta_{k}}{\hbar}}+E_{ik}\big{)}$

We can solve this equation for $P_{n}$ by pre-composing with
$e^{\frac{\alpha_{s}\beta_{i}}{\hbar}+\frac{\xi_{s}\eta_{i}}{\hbar}}$ to
remove the $R[0]_{ij}$ and find:

$(P_{n})_{sk}e^{\frac{\alpha_{s}\beta_{k}}{\hbar}+\frac{\xi_{s}\eta_{k}}{\hbar}}=-E_{ik}//e^{\frac{\alpha_{s}\beta_{i}}{\hbar}+\frac{\xi_{s}\eta_{i}}{\hbar}}$

So finally
$(P_{n})_{sk}=-(E_{ik}//e^{\frac{\alpha_{s}\beta_{i}}{\hbar}+\frac{\xi_{s}\eta_{i}}{\hbar}})e^{-\frac{\alpha_{s}\beta_{k}}{\hbar}-\frac{\xi_{s}\eta_{k}}{\hbar}}$.
By induction on $n$ we see that
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{ij}\in\mathcal{PG}^{\pm}(\\{i,j\\},\emptyset)$.
For this it is important to note that the error $E_{ij}$ has no denominators
because the formula for $R_{ij}$ has none. ∎

Following the pattern of the Drinfeld double construction we set up generating
functions for the Hopf algebra operations of $\mathbb{A},\mathbb{B}$ next.

######  Lemma 38.

(Generating functions for co-product and the antipodes)

  1. 1.

$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}})^{i}_{jk},(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{B}})^{i}_{jk}\in\mathcal{PG}^{+}(\\{i\\},\\{j,k\\})$.

  2. 2.

We have
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{A}})_{i},(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}})_{i},(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\bar{S}_{\mathbb{A}})_{i},(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\bar{S}_{\mathbb{B}})_{i}\in\mathcal{PG}^{+}(\\{i\\},\\{i\\})$
and
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\varepsilon_{\mathbb{A}}^{i}=1=\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\varepsilon_{\mathbb{B}}^{i}=1$.

###### Proof.

We will prove
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}})^{i}_{jk}\in\mathcal{PG}^{+}(\\{i\\},\\{j,k\\})$
and leave the analogous case for
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{B}})^{i}_{jk}$
to the reader. $\boldsymbol{\pi}$ is a Hopf pairing so

$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}})^{i}_{jk}=R_{1k}R_{2j}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{12}_{3}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi_{i3}$

By Lemma 33 (part 2) it suffices to check that the right-hand side at
$\epsilon=0$ can be composed and the result is in $\mathcal{PG}^{\pm}$. We
have

$R_{1k}|_{\epsilon=0}R_{2j}|_{\epsilon=0}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{12}_{3}|_{\epsilon=0}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi_{i3}|_{\epsilon=0}=$
$\langle
e^{\hbar(b_{1}a_{k}+y_{1}x_{k}+b_{2}a_{j}+y_{2}x_{j})+(\beta_{1}+\beta_{2})b_{3}+(\eta_{1}+\eta_{2})y_{3}+\frac{\alpha_{i}\beta_{3}+\xi_{i}\eta_{3}}{\hbar}}\rangle_{1,2,3}=e^{\alpha_{i}(a_{j}+a_{k})+\xi_{i}(x_{j}+x_{k})}$

As this is in $PG^{\pm}$ so is
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}}$. In
fact $\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}}$ is
in $\mathcal{PG}^{+}$ because computing using the definition in terms of
generators shows that negative powers of $\hbar$ do not enter in
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}})^{i}_{jk}=\mathbb{O}^{-1}\mathbf{\Delta}_{\mathbb{A}}(\mathbb{O}e^{\alpha_{i}a_{i}+\xi_{i}x_{i}})$.

From the quasi-triangular axioms it follows that
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}})^{i}_{j}=R^{-1}_{j1}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{1i}$.
By Lemma 33 (part 2) it follows that
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}}$ is in
$\mathcal{PG}^{\pm}$. For this we should check that at $\epsilon=0$ the
composition
$R^{-1}_{j1}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{1i}$ is
well-defined. We get

$R^{-1}_{j1}|_{\epsilon=0}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi^{1i}|_{\epsilon=0}=\langle
e^{-b_{j}a_{1}-B_{j}^{-1}y_{j}x_{1}+\frac{\alpha_{1}\beta_{i}}{\hbar}+\frac{\xi_{1}\eta_{i}}{\hbar}}\rangle_{1}=e^{-b_{j}\beta_{i}-B_{j}^{-1}y_{j}\eta_{i}}$

However $S_{\mathbb{B}}$ was already defined on the generators in the previous
section and its definition does not involve negative powers of $\hbar$.
Therefore the formula
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}})^{i}_{j}=S_{\mathbb{B}}(e^{\beta_{i}b_{j}+\eta_{i}y_{j}})$
shows that in fact
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}})^{i}_{j}\in\mathcal{PG}^{+}(\\{i\\},\\{j\\})$.
Precisely the same argument works for
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{A}}$.

The case of the co-units is clear from the definition. In the remaining cases
of
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\bar{S}_{\mathbb{B}},\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\bar{S}_{\mathbb{A}}$
we apply the next lemma to invert the generating functions for the antipodes.
∎

######  Lemma 39.

(Inverting in $\mathcal{PG}^{+}$)
Suppose $f,g\in\mathcal{PG}^{+}(J,K)$ are such that
$f//g=g//f=\mathrm{id}\mod\epsilon$. Then there exists a unique
$\bar{f}\in\mathcal{PG}^{+}(K,J)$ such that $\bar{f}=g\mod\epsilon$ and
$\bar{f}//f=f//\bar{f}=\mathrm{id}$.

###### Proof.

Uniqueness is a general property of the compositional inverse so we focus on
existence, proving it order by order as usual. We are looking for
$\bar{f}=(1+\sum_{k=0}^{\infty}P_{k}\epsilon^{k})e^{G}$ such that
$\bar{f}//g=\mathrm{id}$ and we have already found $G$ by assumption. Suppose
we found $P_{k}$ for all $k<n$ such that
$(1+\sum_{k=0}^{n-1}P_{k}\epsilon^{k})e^{G}//f=\mathrm{id}+\epsilon^{n}E$ for
some error $E$. If for simplicity we assume that $g$ does not depend on
$\epsilon$ then we can find $P_{n}$ by adding $\epsilon^{n}P_{n}e^{G}//f$ to
both sides, composing with $g$ and taking the coefficient of $\epsilon^{n}$:

$\big{(}1+\sum_{k=0}^{n}P_{k}\epsilon^{k}\big{)}e^{G}//f//g=P_{n}e^{G}+E//g$

By definition of $P_{n}$ this expression should be $0$ so we find
$0=P_{n}e^{G}+E//g$ and so $P_{n}=-e^{-G}(E//g)$. Finally, from
$f,g\in\mathcal{PG}^{+}$ it follows that $\bar{f}\in\mathcal{PG}^{+}$ because
the formula for $E$ transfers all the relevant properties to $P_{n}$. ∎

Turning to the double $\mathbb{D}$ recall $\mathbb{D}$ is a tensor product
itself so as in Subsection 5.3 we use the following notation for elements
$\mathbb{D}^{\otimes J}$. If $j\in J$ and $f\in\mathbb{D}$ then $f_{j}$
denotes the element in $\mathbb{D}^{J}$ that is $1$ except for in factor $j$.
Since $\mathbb{D}=\mathbb{B}\otimes\mathbb{A}$ we use $f_{j_{1}}$ and
$f_{j_{2}}$ for the $\mathbb{B}$ and the $\mathbb{A}$ parts $f_{j}$ so
$f_{j}=f_{j_{1}}f_{j_{2}}$ by definition. We are now ready for the

###### Proof.

(of the main $\mathcal{PG}$ Theorem 34)
Equation (14) defines the multiplication in $\mathbb{D}$ and applying the
functors $\mathcal{O},\mathcal{G}$ turns that equation into an expression for
the generating function
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}})^{ij}_{k}\in\mathcal{C}(\\{i,j\\},\\{k\\})$.
We find

$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}})^{ij}_{k}=(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{B}}^{(2)})^{i_{2}}_{\bar{1}\bar{2}\bar{3}}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}}^{(2)})^{j_{1}}_{123}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{A}}^{-1})_{3}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi_{3,\bar{1}}//\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\pi_{1,\bar{3}}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{B}})^{i_{1},\bar{2}}_{k_{1}}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{A}})^{2,j_{2}}_{k_{2}}$

By Lemmas 37 and 38 all the building blocks of this formula are in
$\mathcal{PG}^{\pm}$. According to Lemma 33 $\mathcal{PG}^{\pm}$ is closed
under composition so
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}}$ is also in
$\mathcal{PG}^{\pm}$. Looking at the definition of
$\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}}$ in terms of
generators and relations it is clear that in fact it is in $\mathcal{PG}^{+}$.

By the same reasoning the generating functions for coproduct, co-unit and
antipode in $\mathbb{D}$ automatically also become morphisms of
$\mathcal{PG}^{+}$ because they can be expressed as a composition of morphisms
in $\mathcal{PG}^{+}$. More precisely we can write their generating series
respectively as

$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{D}})^{i}_{jk}=(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{B}})^{i_{1}}_{j_{1}k_{1}}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}\Delta_{\mathbb{A}})^{i_{2}}_{j_{2}k_{2}}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}})^{j_{1}k_{2}}_{k}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}})^{j_{2}k_{1}}_{k}$
$(\varepsilon_{\mathbb{D}})_{i}=(\varepsilon_{\mathbb{B}})_{i_{1}}(\varepsilon_{\mathbb{A}})_{i_{2}}$
$(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{D}})^{i}_{j}=(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S_{\mathbb{B}})^{i_{1}}_{j_{1}}(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}S^{-1}_{\mathbb{A}})^{i_{2}}_{j_{2}}//(\raisebox{5.0pt}{$\scriptscriptstyle\mathcal{G}$}m_{\mathbb{D}})^{j_{2}j_{1}}_{j}$

Finally the spinner (group-like element)
$\mathbf{C}_{i}=(\mathbf{AB})^{-\frac{1}{2}}$ clearly is mapped into
$\mathcal{PG}$ by applying $\mathbb{O}^{-1}$. From this the ribbon element and
its inverse are easily computed to be in $\mathcal{PG}$ using the formula

$\mathbf{v}_{i}^{-1}=\mathbf{R}_{13}\mathbf{C}_{2}//\mathbf{m}^{123}_{i}$ (18)

The fact that we land in $\mathcal{PG}$ follows from the simple nature of the
formula that is already in canonical order up to powers of $A$. ∎

##  7 From algebra to tangle invariants 2

###  7.1 Rotational tangle diagrams

In this section we introduce a variant of the Morse diagrams of tangles that
are often used in discussing quantum invariants. Such diagrams serve to keep
track of the rotation number of the strands of the diagram. Normally this is
done by marking the local minima and maxima (cups and caps). See for example
the oriented sliced tangle diagrams in section 3.1 of [14] and the discussion
in [18]. We prefer to stick to the rotation numbers themselves and propose the
following definition of rotational tangle diagrams. Our rotational tangle
diagrams are in some ways similar to the rotational virtual knots of [9].

######  Definition 40.

(Rotational tangle diagrams)
The edges of a tangle diagram $D$ are the connected components of the strands
after deleting a small open disk centered at each crossing. The rotation
number $\mathrm{rot}(e)\in\mathbb{Z}$ of edge $e$ is the rotation number of
its tangent vector relative to the vertical888i.e. in the positive $y$
direction of the plane. vector field. A rotational tangle diagram $D$ is a
tangle diagram in the sense of Definition 12 such that the tangent vector at
the endpoints of all the edges is vertical and it is transversal to the
boundary at the end points.

It should be clear that any tangle diagram can be turned into a rotational
tangle diagram by applying local planar isotopies. More specifically one
rotates the crossings and the end-points to make them point upwards as shown
in Figure 6.

Figure 6: Rotating crossings and endpoints to turn a tangle diagram into a
rotational tangle diagram.

A more elaborate example of a rotational tangle is the right hand side of
Figure 7.

Figure 7: Turning the $8_{17}$ tangle diagram from Figure 1 into a rotational
tangle diagram.

As with tangle diagrams we record the simplest rotational tangle diagrams.
Apart from the crossings $X_{ij},\bar{X}_{ij}$ we now also recognize two
important diagrams that look like a $C$. More precisely, $C_{i}$ represents a
crossingless strand labelled $i$ that rotates counter-clockwise. Likewise
$\bar{C}_{i}$ rotates clockwise as shown in Figure 8.

Figure 8: The simplest rotational tangle diagrams. From left to right:
$X_{ij},\bar{X}_{ij}$, $C_{i}$, $\bar{C}_{i}$, $1_{i}$, $1_{ij}$ and
$(1c)_{ij}$.

Using the above simple diagrams we can build more complicated ones by taking
disjoint unions and merging strands. The operations disjoint union and merging
mostly carry over to the rotational setting with some minor restrictions in
the case of merging to take into account the rotation numbers.

######  Definition 41.

(Disjoint union and merging of rotational tangle diagrams)
The disjoint union $DE$ of two rotational tangle diagrams is their disjoint
union as tangle diagrams.

Merging $m^{ij}_{k}$ is defined as for tangle diagrams with the restriction
that the rotation number of the arc $c$ that connects the end of strand $i$ to
the start of strand $j$ has rotation number $0$, see Figure 9

Figure 9: Merging strands $i,j$ in rotational tangle diagram $D$.

######  Definition 42.

(Equivalence of rotational tangle diagrams)
Generate an equivalence relation $"="$ on rotational tangle diagrams by the
following rules, where $D,E,F$ are rotational tangle diagrams:

  1. 1.

$D=E$ if $D,E$ are planar isotopic respecting the orientation, labels on the
strands and rotation numbers of the edges.

  2. 2.

If $D=E$ then $DF=EF$.

  3. 3.

If $D=E$ then $D//m^{ij}_{k}=E//m^{ij}_{k}$, provided both make sense.

  4. 4.

$D=E$ if $D$ and $E$ appear in one of the rotational Reidemeister equalities
shown in Figure 10.

Figure 10: The rotational Reidemeister moves, together with their algebraic
description.

######  Lemma 43.

(Tangles inject into rotational tangles)
For any tangle diagram $D$ there exists a rotational tangle diagram
$D^{\prime}$ is planar isotopic to it.

Moreover if two tangle diagrams $D,E$ are equivalent then any of the
corresponding rotational tangle diagrams $D^{\prime},E^{\prime}$ are
equivalent (as rotational tangles) up to change of framing and rotation number
at the end points.

###### Proof.

Given tangle diagram $D$ a rotational tangle diagram planar isotopic to it is
obtained by locally rotating the crossings and endpoints so they point
upwards.

Any of the equivalences generating equivalence of tangle diagrams can be
rotated similarly to yield a rotational equivalence. The exceptions are the
planar isotopies where one needs the swirls and the Reidemeister 1. The latter
means we should ignore the framing. The rotation number at the endpoints is
also arbitrary. ∎

An important reason for keeping track of the framing of our strands is that
doubling a strand is a well-defined operation. Likewise an important reason
for keeping track of the rotation numbers is to make sure strand reversal is
well-defined and has good properties that reflect the properties of Hopf
algebras as we will see in the next subsection.

######  Definition 44.

(Strand doubling, reversal and deletion)
Define the operations $\varepsilon,\Delta,S,\bar{S}$ on rotational tangle
diagrams as follows. Suppose $D$ is a rotational tangle diagram that has a
strand labelled $i$ and there are no strands labeled $r,\ell$.

  1. 1.

Define $D//\varepsilon^{i}$ to be the diagram obtained from $D$ by deleting
strand $i$.

  2. 2.

Define $D//\Delta^{i}_{r,\ell}$ to be the diagram obtained from $D$ by
choosing a small tubular neighborhood $N$ of strand $i$ and replacing strand
$i$ by the two sides of $N$ that run parallel to $i$. The new strand to the
left of $i$ is called $\ell$ and the other is called $r$. The newly created
crossings involving strands $r,\ell$ should all have the same sign as the
corresponding crossing on strand $i$.

  3. 3.

Define $D//S_{i}$ to be the diagram obtained from $D$ by rotating the
endpoints of strand $i$ half a turn in the clock-wise direction and then
reversing the orientation of strand $i$. To get a proper rotational diagram
the crossings need to be rotated upwards as in Figure 8.

  4. 4.

Define $D//\bar{S}_{i}$ to be the same as $D//S_{i}$ except that the ends are
to be rotated half a turn in the counter-clockwise direction.

Figure 11: A generic rotational tangle diagram $D$ (middle) together with the
effect of the operations $\epsilon,\Delta,S,\bar{S}$ that delete, double and
reverse strand $i$. Notice that $S$ looks like an $S$.

Notice how the effect of the operation $S$ on a straight strand turns it into
an ’S’. Less fortunate is that $\Delta^{i}_{r\ell}$ places the new strand $r$
to the right and the $\ell$ strand to the left relative to the framing and the
orientation of strand $i$.

As an illustration of the use of these properties we show how to build the
(right-handed) Whitehead double of any long knot diagram. Using the notation
$v_{i}=\bar{X}_{13}C_{2}//m^{123}_{i}$ for the negative kink (Reidemeister $1$
curl) we set

$W^{i}_{0}=\Delta^{i}_{jk}//S_{j}X_{18}X_{62}v_{3}v_{4}\bar{C}_{7}C_{5}//m^{1234j5678k}_{0}$
(19)

Figure 12: Left: taking the (right-handed) Whitehead double of strand $i$ in
diagram $D$. Right: the Whitehead double of the figure eight knot.

As illustrated in Figure 12, for any $0$-framed rotational long knot diagram
$D_{i}$ with strand labelled $i$, a $0$-framed rotational diagram of the
Whitehead double is $D_{i}//W^{i}_{0}$.

As a more elaborate illustration we discuss Seifert surfaces in a way similar
to [6].

######  Lemma 45.

(Seifert surface criterion)
Define

$\mathcal{B}^{ij}_{k}=C_{3}C_{4}\Delta^{i}_{r_{1}\ell_{1}}\Delta^{j}_{r_{2}\ell_{2}}//\bar{S}_{r_{1}}//S_{r_{2}}//m^{\ell_{1}r_{2}34r_{1}\ell_{2}}_{k}$
(20)

If rotational long knot diagram $K$ represents the boundary of a genus $g$
Seifert surface then there exists a $0$-framed rotational tangle diagram $L$
with $2g$ strands named $1,\dots 2g$ such that

$K_{1}=L//_{j=1}^{g}\mathcal{B}^{2j-1,2j}_{j}//m^{12\dots g}_{1}$ (21)

###### Proof.

Given a Seifert surface for a knot $K$ recall (e.g. [3] p.107) that can we
bring it into band form where the surface looks like a disk with $2g$ bands
attached as shown in Figure 13. The bands may be described by a blackboard
framed tangle $L$ consisting of the cores of these bands.

Figure 13: A schematic picture for the Seifert surface of knot $K$ as a disk
with $2g$ bands attached. In red we show the cores of the bands that make up
the tangle $L$.

To obtain the knot $K$ from the tangle we need to thicken each strand using
the $\Delta$ operation and then fix the orientation using $S,\bar{S}$. We need
to correct the half turns coming from the $S$ and one way to do this is
include two copies of $C$ in the middle as shown in Figure 14 in the special
case of the figure eight knot. Connecting the strands in the correct way leads
us to consider the operation $\mathcal{B}^{ij}_{k}$ defined above. ∎

Figure 14: Left: the $2g$-component tangle encoding the bands of a Seifert
surface for the figure eight knot. Here the genus $g=1$ and thickening the
components using $\mathcal{B}^{ij}_{k}$ reconstructs the boundary of the
surface shown in the middle. Notice the cups and caps are organised so as to
reflect the $S,\bar{S}$ and $C^{2}$ in the formula. Right: The formula of
$\mathcal{B}^{ij}_{k}$ is illustrated more explicitly with the cups and caps
belonging to the $S,\bar{S}$ shown in red and the two $C^{\prime}s$ in green.

###  7.2 Universal invariant of rotational tangles

In this section we sketch how a ribbon Hopf algebra gives rise to a knot
invariant and more generally an invariant of tangles known as the universal
invariant (corresponding to that algebra). A general reference for what
follows is [14]. Our purpose here is just to establish notation for the
universal invariant. When we speak about knots we will always mean one-strand
rotational tangles (long knots).

######  Definition 46.

Suppose $A$ is a ribbon Hopf algebra $A$ with unit $\mathbf{1}$ and
multiplication $\mathbf{m}^{ij}_{k}:A^{\otimes\\{i,j\\}}\to
A^{\otimes\\{k\\}}$ with universal $R$-matrix $\mathbf{R}_{ij}$ and spinner
$\mathbf{C}$ as in subsection 5.2. For a rotational tangle diagram $D$ whose
strands are labeled by set $L$ define $\mathbf{Z}_{A}(D)\in A^{\otimes L}$ by
the following rules.

  1. 1.

If $D$ does not have crossings, $\mathbf{Z}_{A}(D)=\prod_{\ell\in
L}\mathbf{C}_{\ell}^{\mathrm{rot}(\ell)}$. (crossingless diagrams).

  2. 2.

$\mathbf{Z}_{A}(X_{ij}^{\pm 1})=\mathbf{R}^{\pm 1}_{ij}$ (value of the
crossings).

  3. 3.

If diagram $E$ is labelled by set $M$ then
$\mathbf{Z}_{A}(DE)=\mathbf{Z}_{A}(D)\otimes\mathbf{Z}_{A}(E)\in A^{\otimes
L\sqcup M}$
(disjoint union is tensor product).

  4. 4.

$\mathbf{Z}_{A}(D//m^{ij}_{k})=\mathbf{Z}_{A}(D)//\mathbf{m}^{ij}_{k}$
(merging is multiplication).