hep-th/??????

CECS-PHY-10/08

Kauffman Knot Invariant from
SO(N) or Sp(N) Chern-Simons theory
and the Potts Model
Marco<EMAIL_ADDRESS>

Instituto de Física,
Pontificia Universidad Católica de Valparaíso
and
Centro de Estudios Científicos (CECS), Valdivia,
Chile

Abstract

The expectation value of Wilson loop operators in three-dimensional SO(N)
Chern-Simons gauge theory gives a known knot invariant: the Kauffman
polynomial. Here this result is derived, at the first order, via a simple
variational method. With the same procedure the skein relation for Sp(N) are
also obtained. Jones polynomial arises as special cases: Sp(2), SO(-2) and
SL(2,$\mathbb{R}$).
These results are confirmed and extended up to the second order, by means of
perturbation theory, which moreover let us establish a duality relation
between SO($\pm$N) and Sp($\mp$N) invariants.
A correspondence between the firsts orders in perturbation theory of SO(-2),
Sp(2) or SU(2) Chern-Simons quantum holonomies and the partition function of
the $Q=4$ Potts Model is built.

##  1 Introduction

In a milestone work [1] Witten realised that the expectation value of a Wilson
loop, computed with a three-dimensional Chern-Simons action measure, was a
knot invariant. This is due to the fact that the Wilson loops are observables
for Chern-Simons theories, having therefore diffeomorphism invariant
expectation values. More in general this feature stems from the property that
such a quantum field theory manifests general covariance, which in turn is a
consequence of the metric independent structure: any physical quantity
computed in this framework is a topological invariant.
In practise, for SU(N) Chern-Simons field theory, the resulting knot invariant
is the HOMFLY polynomial, which in particular specialises into the Jones
polynomial in the case of SU(2). These outcomes were derived through both
conformal field theory (as in [1]) or perturbative quantum field theory (see
for instance [2]). But a simpler heuristic derivation was proposed in [3] and
[4] (for reviews see also [5] and [6]), at least up to the first order in the
inverse coupling constant of the theory. It is based on a variational
approach: it studies the behaviour in the expectation value of the Wilson loop
when one performs small geometric deformation.
In the conformal field theory scheme similar results have been found in [9],
[10] and [11] for several other groups: SO(N), Sp(N), SU(n$|$m) and
OSp(m$|$2n).
It would be interesting to test whether the variational procedure, which is
expressly realised to reproduce the HOMFLY polynomial from SU(N) gauge theory,
may apply also in different contexts. In section 3 are studied the SO(N),
SL(N,$\mathbb{R}$) and Sp(N) cases.
The results obtained are moreover analysed in section 4 by means of the more
rigorous standard perturbation theory and extended up to the subsequent order,
the second.
Finally in section 5 we try to interpret these results from the statistical
mechanic point of view, trying to connect the holonomies’s first order
expansion to one of the more famous lattice statistical system: the Q-Potts
Model111We are referring to the standard two dimensional Potts model, not to
some variant with multiple Boltzmann weights, which in much literature are
misleading called in same way.; which at the moment remains unsolved apart for
its easiest personification when Q=2, the Ising model. We start (section 2)
introducing the notation and summarising the fundamental properties of Chern-
Simons theory and Kauffman polynomial that are useful in derivation of skein
relations.

##  2 Chern-Simons theory and Kauffman polynomial

Let’s consider a Chern-Simons theory for a gauge field connection one-form
$A=A^{a}_{\ \mu}(x)T^{a}dx^{\mu}$ valued in a generic semi-simple Lie algebra
$\mathfrak{g}$, with action:

$\mathcal{L_{CS}}[A]=\frac{k}{4\pi}\int_{\mathcal{M}^{3}}d^{3}x\
\frac{\epsilon^{\mu\nu\rho}}{2}\ \left(A^{a}_{\ \mu}\partial_{\nu}A^{a}_{\
\rho}-\frac{1}{3}A^{a}_{\ \mu}A^{b}_{\ \nu}A^{c}_{\ \rho}f^{abc}\right)$

where $\mathcal{M}^{3}$ is a compact three-dimensional manifold whose
coordinates are labelled by Greek letters ($\mu,\nu,\rho,...$); while the
internal group indices will be denoted by Latin letters ($a,b,c,...$). The Lie
algebra is spanned by generators $T^{a},T^{b},\dots$, obeying the commutation
relations $[T^{a},T^{b}]=if^{abc}T^{c}$ and normalised as follows: ${\rm
Tr}\;(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$.

This action got several notable properties: $(i)$ it changes by $2\pi kn_{g}$
under a gauge transformation $A_{\mu}\rightsquigarrow
A^{\prime}_{\mu}=g^{-1}A_{\mu}g-ig^{-1}(\partial_{\mu}g)$ ($n_{g}$ is the
degree of the mapping $g:\mathcal{M}^{3}\rightarrow\mathcal{G}$); thus,
$\forall\ k\in\mathbb{Z}$, $\mathrm{exp}(i\mathcal{L_{CS}})$ is a complete
gauge invariant quantity that will play the rôle of the path integral measure.
$(ii)$ The curvature of the gauge field at the point $x\in\mathcal{M}^{3}$ is
given by:

$F^{a}_{\ \mu\nu}(x)=\frac{4\pi}{k}\epsilon_{\mu\nu\lambda}\
\frac{\delta\mathcal{L_{CS}}[A(x)]}{\delta A^{a}_{\ \lambda}(x)}$

We will interested in computing expectation values $\langle W(\gamma)\rangle$
for Wilson loops $W_{\gamma}[A]$ along closed paths $\gamma$, that in fact may
be thought as a knot on $\mathcal{M}^{3}$, defined as follows:

$\displaystyle W_{\gamma}[A]$ $\displaystyle=$ $\displaystyle{\rm
Tr}\;\Big{[}\mathrm{P}\
\mathrm{exp}\Big{(}i\oint_{\gamma}A_{\mu}dx^{\mu}\Big{)}\Big{]}$
$\displaystyle\langle W(\gamma)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{-1}\ \int\mathscr{D}A\ \mathrm{exp}\left(i\
\mathcal{L_{CS}}[A]\right)\ W_{\gamma}[A]$

In this notation $\gamma$ represents both common knots
$\gamma(t):I\rightarrow\mathcal{M}^{3}$ and n-component knots, also called
knot-links,
$\gamma(t_{1},t_{2},\dots,t_{n})=(\gamma_{1}(t_{1}),\gamma_{2}(t_{2}),\dots,\gamma_{n}(t_{n})):I_{1}\times
I_{2}\times\dots I_{n}\rightarrow\mathcal{M}^{3}$. In the latter case $\langle
W(\gamma)\rangle=\langle W(\gamma_{1})W(\gamma_{2})\dots
W(\gamma_{n})\rangle$. Without losing generality one may think the compact
interval $I_{i}=[0,1]$ and $\gamma(0)=\gamma(1)$ in order to have closed
paths. The fact that Chern-Simons action is independent of the particular
choice of a metric on the three-manifold suggests that the Wilson loop
expectation values may capture some invariant or topological characteristic of
the system’s geometry: either that of the knots or of the manifold itself.

Now we introduce the Kauffman polynomial which is a regular isotopy invariant
of knots and, if suitably normalised, becomes an ambient isotopy invariant.
Actually we will deal with its equivalent Dubrovnik version. To each knot-link
there is associated a finite Laurent polynomial $D_{K}=D_{K}(a,z)$ of two
variables with integer coefficients, such that if $K_{1}\sim K_{2}$, then
$D_{K_{1}}=D_{K_{2}}$ (while the reverse is not necessary true). The
polynomial can be constructed, as in [7] or [14], by the following
rules222Sometimes, as in [5], can be found a different normalisation for
$D_{K}$: $iii)^{\prime}\ \ D(\bigcirc)=1+\frac{a-a^{-1}}{z}$; in our notation
$1+\frac{a-a^{-1}}{z}$ will result the $\langle\bigcirc\rangle$ ’s
normalisation. (see figure 1 for notation, $\bigcirc$ stands for the unknotted
circle):

$\displaystyle i)$ $\displaystyle\qquad D(L_{+})-D(L_{-})=z\
[D(L_{0})-D(L_{\infty})]$ $\displaystyle ii)$ $\displaystyle\qquad
D(\hat{L}_{\pm})=a^{\pm}D(\hat{L}_{0})$ $\displaystyle iii)$
$\displaystyle\qquad D(\bigcirc)=1$

Figure 1: Different crossing configurations involved in the skein relations.
Dealing with unoriented links, arrows can be ignored because they carry no
sensitive information.

In $i)$ and $ii)$ the small diagrams $\\{L_{k}\\}_{k=\pm,0,\infty}$ stand for
larger link diagrams that differ only as indicated by the smaller ones.
Starting from any knot-links K and using recursively Reidemeister moves and
the skein relations (2) at each diagram’s crossing, one can obtain uniquely
its regular isotopy invariant $D_{K}(a,z)$. It is possible to normalise
$D_{K}$ by a factor that take into account also eventual contributions of
twists. For this purpose is used the writhe $w(K)=\sum_{p}\epsilon(p)$, where
$p$ runs over all crossing in $K$ and $\epsilon(L_{\pm})=\pm 1$ is the sign of
the type of crossing. So finally we are able to define a genuine ambient
isotopy invariant: the normalised Kauffman-Dubrovnik polynomial333While
$D_{k}$ is defined for unoriented knots, to calculate the writhe in $Y_{K}$
one needs to define an orientation. At the end the orientation does not affect
the result for knots but it affects the invariant polynomial in case of proper
links. Thus $Y_{K}$ is said to be defined for semi-oriented knot-links.:

$Y_{K}(a,z)=(a)^{-w(K)}D_{K}(a,z)$

##  3 Variational derivation of the skein relation

It’s well known (see [5] for details) that the Wilson loops satisfy the
following differential equations:

$\displaystyle\delta_{A}W_{\gamma}[A]$ $\displaystyle=$
$\displaystyle\frac{\delta W_{\gamma}[A]}{\delta A^{a}_{\ \mu}(x)}=i\ T^{a}\
dx^{\mu}\ W_{\gamma}[A]$ $\displaystyle\delta_{\gamma_{x}}W_{\gamma}[A]$
$\displaystyle=$ $\displaystyle iF^{a}_{\ \mu\nu}T^{a}\
dx^{\mu}dx^{\nu}W_{\gamma}[A]$

where $\delta_{\gamma_{x}}$ is the variation corresponding to an infinitesimal
deformation of the loop $\gamma$ in the neighbourhoods of a point $x$. It’s
then possible to compute this variation for an expectation value of a Wilson
line along a knotted path $\gamma$ and to use it to obtain a formula for the
switching identity $\langle W(\hat{L}_{+})\rangle-\langle
W(\hat{L}_{-})\rangle$ as444Proposition 17.4 and theorem 17.5 of [5]. follows:

$\delta_{\gamma_{x}}\langle W(\gamma)\rangle=-\frac{4\pi
i}{k}\frac{1}{\mathcal{Z}}\int\mathscr{D}A\ \mathrm{exp}\ \left(i\
\mathcal{L_{CS}}[A]\right)\
\left[\epsilon_{\mu\nu\lambda}dx^{\mu}dx^{\nu}dy^{\lambda}\right][\sum_{a}T^{a}T^{a}]W_{\gamma}[A]$
(3.1)

Note that studying the formal properties of this integral three assumptions
are always used: $i)$ the limits of differentiation and integration commute:
$\delta_{\gamma_{x}}\langle
W_{\gamma}[A]\rangle=\langle\delta_{\gamma_{x}}W_{\gamma}[A]\rangle$; $ii)$
integrating by parts it’s possible to discard the boundary term; $iii)$ the
existence of an appropriate functional measure on this moduli space.
From the previous equation one is able to write the switching identity
$\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle$. The quantity
$\left[\epsilon_{\mu\nu\lambda}dx^{\mu}dx^{\nu}dy^{\lambda}\right]$ is
dimensionless and, whether properly normalised, can be thought -1,0 or 1. Then
(3.1) has a standard interpretation (we follow [5]) if one calls the operator,
which in some sense enclose the loop’s small deformation,
$C=\sum_{a}T^{a}T^{a}$:

$\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle=-\frac{4\pi
i}{k}\langle C\ W(\gamma)\rangle$ (3.2)

Graphically $\langle C\ W(\gamma)\rangle$ is represented in the l.h.s of
figure’s 2 equation. Note that the sign is a convention which may be reversed
exchanging $\hat{L}_{+}\leftrightarrow\hat{L}_{-}$.
Till this point the whole model has been valid for a generic gauge group
$\mathcal{G}$. In particular was successfully used in the literature to
reproduce the Witten’s result for HOMFLY polynomials from the SU(N) group.
Instead in this paper we specialise our study to two particular algebras which
have simple Fierz identities: the ones associated to the orthogonal group
SO(N) and the symplectic group Sp(N), for a generic N.

###  3.1 SO(N) and Kauffman polynomial

Here the features of the algebra under consideration begin to play an
important rôle. In fact to evaluate the operator $C$ one needs to use the
Fierz identity; in particular we have for SO(N) in the fundamental
representation (in [8] Fierz identities are presented for almost all semi-
simple Lie groups):

$\sum_{a}(T^{a})^{i}_{\ j}(T^{a})^{k}_{\ l}=\frac{1}{4}\left(\delta^{i}_{\
l}\delta^{k}_{\ j}-\delta^{ik}\delta_{jl}\right)$

This expression in the Baxter’s abstract tensor notation (see [5]) reads as
the diagrammatic relation drawn in figure 2.

Figure 2: Abstract diagrammatic representation of Fierz identity for SO(N)

Hence, substituting in (3.2) the Fierz identity we have:

$\langle W(L_{+})\rangle-\langle W(L_{-})\rangle=-\frac{\pi
i}{k}\big{[}\langle W(L_{0})\rangle-\langle W(L_{\infty})\rangle\big{]}$ (3.3)

To get in touch with the known results, one has to take the limit of $k>>1$,
namely the analogous of the first order perturbation expansion, thus the
previous expression reduces to:

$\langle W(L_{+})\rangle-\langle W(L_{-})\rangle=\big{(}q-q^{-1}\big{)}\
\big{[}\langle W(L_{0})\rangle-\langle W(L_{\infty})\rangle\big{]}$

These are exactly the skein relations that are found by means of the original
Witten’s method based on conformal field theory arguments (see [9] and [10]),
once $q:=\mathrm{exp}(-\frac{\pi i}{2k})$ is defined555[10] uses a different
killing metric normalisation for the Lie algebra generators; in order to
compare with it one has to define a slightly different
$q:=\textrm{exp}(-\frac{\pi i}{k})$. [9] uses an inverse definition of the
writhe and of the crossing diagrams, so what they call $\alpha=a^{-1}$ and
their $q$ is our $q^{-1}$.. So is not difficult to see that $D_{K}=\langle
W(K)\rangle/\langle W(\bigcirc)\rangle$ fulfils the definition of Dubrovnik
polynomial (normalised as in [7] and [14]666Clearly if write-normalised by a
factor $a^{-w(K)}$ (where $w(L_{\pm})=\pm 1$) $D_{K}(a,z)$ became an ambient
isotopy invariant.), with $z=(q-q^{-1})$. The only thing that remains to fix
is the value of $a$ such that $\langle W(\hat{L}_{+})\rangle=a\langle
W(\hat{L}_{0})\rangle$. This can be done considering the closure of the path
in the skein relation (3.3), as shown in the figure below:

Figure 3: Diagrammatic closure of the SO(N) skein relation (3.3)

$\displaystyle\langle W(\hat{L}_{+})\rangle-\langle W(\hat{L}_{-})\rangle$
$\displaystyle=$ $\displaystyle-\frac{\pi i}{k}\big{[}\langle W(\bigcirc\
\hat{L}_{0})\rangle-\langle W(\hat{L}_{0})\rangle\big{]}$ $\displaystyle
a\langle W(\hat{L}_{0})\rangle-a^{-1}\langle W(\hat{L}_{0})\rangle$
$\displaystyle=$ $\displaystyle-\frac{\pi i}{k}\big{[}(N-1)\langle
W(\hat{L}_{0})\rangle\big{]}$ (3.4)

Solutions for (3.1) are $a=q^{N-1}$ or $a=-q^{1-N}$, which however gives rise
at an equivalent $D_{K}$ polynomials777Just redefine
$q\rightarrow\tilde{q}=-q^{-1}$ to verify the second root branch redundancy..
The factor $N$ comes from the diagrammatic tensor interpretation of the unknot
circle, that is $\delta_{i}^{\ i}=N$. It’s worth to observe that these
Dubrovnik-Kauffman polynomials $D_{K}(a=-q^{1-N},z=q-q^{-1})$ do not run out
all the original ones, but constitute a smaller subset depending on the fact
that $a$ assumes only discrete values depending on $N$ (which generally is
thought in $\mathbb{N}$).
The consistency check up to the $1/k$ order proposed in [3] is intrinsically
satisfied using the quadratic Casimir operator of
$\mathfrak{so}(N):\mathbbm{1}(N-1)/4$. Moreover the variational first order
approach, can be generalised to subsequents orders with the same arguments
presented in [12] and [13] for SU(N) groups. But we will prefer explore the
subsequent order of the expansion (see section 4) through a different method
based on the standard quantum field theory of perturbations.
Finally note that the original Jones polynomial
$a^{-w(K)}D_{K}(\bar{a}=-q^{3},\bar{z}=q-q^{-1})$ is not included in this sub-
class of Kauffman polynomial, unless choosing unconventionally $N=-2$ (once
the polynomial is analytic continued for all integers values of N).
Negative dimensions group theory is a powerful technique, first introduced by
Penrose, to calculate algebraic invariants (see [15], [16] and [17]). In that
case it relates the Casimirs and Young tableau of SO(-2) to the ones of Sp(2).
Some speculation about this possibility are done in the next subsection, while
a more rigorous treatment is done on section 4.
One may be puzzled not to come across Jones polynomial for the SO(3) group
which is locally isomorphic to SU(2) where this relation holds. The reason for
this mismatch is based on the fact that in this context, more than groups
similarities, the Lie algebras invariants play a key rôle.
Actually, as also for SL(2,$\mathbb{R}$) generators the same SU(2) Fierz
identity for the $C$ operator holds, Jones polynomial can be recovered with
the same procedure of [3]. It is not surprising because
$\mathfrak{sl}(2,\mathbb{R})$ is the real split form of the $A_{1}$ algebra
(known also as the $\mathfrak{sl}(2,\mathbb{C})$ algebra by an abuse of
notation), while $\mathfrak{su}(2)$ is the real compact one.

###  3.2 Sp(N) skein relations and Jones Polynomial for Sp(2)

In this section we consider the Symplectic group Sp(N), for even N; apart from
the relation with SO(-N) it is an interesting case for itself. Its Fierz
identity (see again [8]) for the generators in the fundamental representation
is:

$\sum_{a}(T_{a})^{i}_{\ j}(T_{a})^{k}_{\ l}=\frac{1}{4}\left(\delta^{i}_{\
l}\delta^{k}_{\ j}+f^{ik}f_{jl}\right)$

where $f^{ij}=-f^{ji}\ ,\ f^{ij}f_{jk}=\delta^{i}_{\ k}$. As the fundamental
representation of this group is pseudoreal, unlike SO(N), the orientation
should not be neglected as it is shown in figure 4.888In [10] another approach
(which has the advantage that leaves the Wilson lines unoriented) is also
presented, but not preferred as requires the specific choice of a ”time”
direction, which breaks the topological invariance because it is no longer
possible to freely rotate the Wilson lines. Plugging this Fierz identity for
Sp(N) into eq. (3.2) one fits the same skein relation of [10] which is
obtained by a totally different approach.999We refer to the one drawn in
figure 17 of [10]

Figure 4: Fierz identity for Sp(N), dots represent points where orientations
of the line change.

There is a particular case where those computation are easily101010Even
without the oriented diagram notation which is unnecessary heavy for Sp(2).
One might work, in a complete compatible way, with the arrowed diagrams but
paying the price of redefining appropriate oriented Reidemeister moves and
oriented Kauffman state bracket as described in cap $6^{0}$ of [5] and [10].
carried on till get its knot invariant: N=2, just the one suspected to be
related to the Jones polynomial, as we saw in section 3.1. In fact for Sp(2)
the antisymmetric matrix $f^{ij}$ may be straight interpreted, without losing
generality, as the Levi-Civita tensor $\epsilon^{ij}$ and its inverse
$f_{ij}=-\epsilon_{ij}$ . Hence the algebraic (eq. (3.5)) and diagrammatic
(fig. 5) representations of the C operator appear respectively as follows:

$\displaystyle\sum_{a}(T_{a})^{i}_{\ j}(T_{a})^{k}_{\
l}=\frac{1}{4}\left(\delta^{i}_{\ l}\delta^{k}_{\
j}-\epsilon^{ik}\epsilon_{jl}\right)=\frac{1}{4}\left(2\delta^{i}_{\
l}\delta^{k}_{\ j}-\delta^{i}_{\ j}\delta^{k}_{\ l}\right)$ (3.5)

Figure 5: Diagrammatic representation of Fierz identity for Sp(2)

Now substituting the Fierz identity for Sp(2) into (3.2) we have:

$\displaystyle\langle W(L_{+})\rangle-\langle W(L_{-})\rangle$
$\displaystyle=$ $\displaystyle-\frac{2\pi i}{k}\langle
W(L_{0})\rangle+\frac{\pi i}{2k}\langle W(L_{+})\rangle+\frac{\pi
i}{2k}\langle W(L_{-})\rangle$ $\displaystyle\left(1-\frac{\pi
i}{2k}\right)\langle W(L_{+})\rangle$ $\displaystyle-$
$\displaystyle\left(1+\frac{\pi i}{2k}\right)\langle W(L_{-})=-\frac{2\pi
i}{k}\langle W(L_{0})\rangle$ $\displaystyle q\langle
W(L_{+})\rangle-q^{-1}\langle W(L_{-})\rangle$ $\displaystyle=$
$\displaystyle\tilde{z}\langle W(L_{0})\rangle$

Where $q$ is the same of section 3.1, while it is defined
$\tilde{z}:=-\frac{2\pi i}{k}=x-x^{-1}$ if we call $x:=\textrm{exp}(-\frac{\pi
i}{k})$. Again we are considering at this stage $k>>1$, i.e these equalities
hold up to first order in the inverse coupling constant of the theory111111The
first order consistency check proposed in [3] is trivially satisfied using,
this time, the quadratic Casimir operator of $\mathfrak{sp}(2):\
3\mathbbm{1}/4$. Closing the path in the previous skein relation as done for
SO(N) we will be able to get a constraint that reduces one variable
dependence:

$\displaystyle q\langle W(\hat{L}_{+})\rangle-q^{-1}\langle
W(\hat{L}_{-})\rangle$ $\displaystyle=$ $\displaystyle\tilde{z}\langle
W(\hat{L}_{0}\ \bigcirc)\rangle$ $\displaystyle aq\langle
W(\hat{L}_{0})\rangle-a^{-1}q^{-1}\langle W(\hat{L}_{0})\rangle$
$\displaystyle=$ $\displaystyle x^{2}-x^{-2}\langle W(\hat{L}_{0})\rangle$
$\displaystyle\Longrightarrow\qquad aq$ $\displaystyle=$ $\displaystyle x^{2}$

As before the second root $aq=-x^{-2}$ leads exactly to the same results. So
at large values of $k$ for a normalised (to be a) expectation value
$P(K)=a^{-w(K)}\langle W(K)\rangle/\langle W(\bigcirc)\rangle$ the original
one variable Jones polynomial follows directly:

$x^{2}P(L_{+})-x^{-2}P(L_{-})=(x-x^{-1})P(L_{0})$

So actually the estimation suggested by negative dimension group theory seems
to work reliably. As it’s here proved the Sp(2) Chern-Simons expectation
values of a Wilson knot-link gives the Jones polynomial invariant for the same
link.

##  4 Perturbative Quantum Field approach

It’s worth analysing the heuristic previous section’s results in a more
carefully way. We opt for the standard quantum field theory of perturbation as
developed for the SU(N) group in [2], which maybe got the disadvantage of
being less qualitative from a geometrical point of view but got the benefit of
being more analytically quantitative. The fact of being, in principle, a
different approach also adds some guaranties on the consistency of the check.
Not least this method let us push the expansion, in the inverse coupling
constant $k$, to one order further.
Note that for this procedure a framing of the knot is needed; in this paper is
always used the _vertical frame_ defined as the one that got linking number
equal to the writhe of the knot $\varphi_{f}(K)=w(K)$. Framed knots can be
thought as bands, so in this picture a writhe for a knot represents a band
twist. As Kauffman polynomial are regular isotopy invariant, twisted bands are
the most suitable objects to be described with. The expectation value for the
Wilson loop computed along a vertical framed, m-component
($C_{1},C_{2},...,C_{m}$) knot-link $K$ in a Chern-Simons theory for a generic
semisimple group $\mathcal{G}$ is given at second order by:

$\displaystyle\langle W(K)\rangle$ $\displaystyle=$
$\displaystyle\Big{(}\prod_{k=1}^{m}\textrm{dim}\
T_{k}\Big{)}\Big{\\{}1-i\Big{(}\frac{2\pi}{k}\Big{)}\sum_{k=1}^{m}c_{2}(T_{k})\varphi_{f}(C_{k})$
$\displaystyle-$ $\displaystyle\Big{(}\frac{2\pi}{k}\Big{)}^{2}\
\sum_{k=i}^{m}\Big{[}\frac{1}{2}c_{2}^{2}(T_{k})\varphi^{2}_{f}(C_{k})-c_{v}c_{2}(T_{k})\rho(C_{k})\Big{]}$
$\displaystyle-$
$\displaystyle\Big{(}\frac{2\pi}{k}\Big{)}^{2}\sum_{k\neq\ell}c_{2}(T_{k})c_{2}(T_{\ell})\Big{[}\varphi_{f}(C_{k})\varphi_{f}(C_{\ell})+\frac{\chi^{2}(C_{k},C_{\ell})}{\textrm{dim}\
\mathcal{G}}\Big{]}+O\Big{(}\frac{1}{k^{3}}\Big{)}\Big{\\}}$

where $T$ stands for the fundamental representation, $\chi(C_{k},C_{\ell})$ is
the Gauss linking number between the two curves $C_{k}$ and $C_{\ell}$,
$\big{(}c_{2}(T)\big{)}_{i}^{\ j}=\sum_{a}(T^{a})_{i}^{\ k}\ (T^{a})_{k}^{\
j}$ is the quadratic Casimir in the fundamental representation, $c_{v}$ the
quadratic Casimir in the adjoint representation, $\rho(C)$ is an ambient
isotopy invariant characteristic of the knot under consideration. $\rho(C)$
represents the second coefficient of the Alexander-Conway polynomial and is
related with Arf- and Casson-invariants; in practise it is not easy to compute
apart from small knots.
Our aim is now, with the help of (4), to find the value of $a$ appearing in
(2-$ii$) in terms of its expansion in $(2\pi/k)$. The effect of changing the
frame of a link component $C_{i}$ by $\Delta\varphi_{f}(C_{i})=\Delta
w(C_{i})=\pm 1$ (or adding a twist in the band picture) reflects in the entire
Wilson loop expectantion value by:

$\langle W(K_{\varphi\pm 1})\rangle=\alpha^{(\pm)}\langle
W(K_{\varphi})\rangle$ $\alpha^{(\pm)}=1\mp
i\Big{(}\frac{2\pi}{k}\Big{)}c_{2}(T)-\frac{1}{2}\Big{(}\frac{2\pi}{k}\Big{)}^{2}c_{2}^{2}(T)+O\Big{(}\frac{1}{k^{3}}\Big{)}$
(4.2)

So we find $a^{\pm 1}=\alpha^{(\pm)}$, taking into account $D_{K}=\langle
W(K)\rangle/\langle W(\bigcirc)\rangle$ as previously defined on section 3.1.
While (2-$iii$) is trivially satisfied, is possible to extract the value of
$z$ from (2-$i$), for instance applying it to the Hopf-link $\mathcal{HL}$.

Figure 6: Skein relation 2-$i)$ applied to the the upper $\mathcal{HL}$
crossing

That is closing the skein relation (2)-$i$ as shown above one gets the
following expression:

$D_{\mathcal{HL}}-D_{\bigcirc\bigcirc}=z(a-a^{-1})D_{\bigcirc}$

written in term of relatively easy objects that can be computed directly from
(4), using as in [2], $\rho(\bigcirc)=-1/12$:

$\displaystyle D_{\bigcirc\bigcirc}$ $\displaystyle=$ $\displaystyle
N\left[1-\frac{1}{12}\left(\frac{2\pi}{k}\right)^{2}c_{v}c_{2}(T)+O\left(\frac{1}{k^{3}}\right)\right]$
(4.3) $\displaystyle D_{\mathcal{HL}}$ $\displaystyle=$ $\displaystyle
N\left[1-\frac{1}{12}\left(\frac{2\pi}{k}\right)^{2}c_{v}c_{2}(T)-\left(\frac{2\pi}{k}\right)^{2}c_{2}^{2}(T)\frac{2}{\textrm{dim}\mathcal{G}}+O\left(\frac{1}{k^{3}}\right)\right]$

An alternative way to find $z$ is imposing the equality between Kauffman
$D_{K}(a,z)$ polynomials obtained from the skein relations (2) with the
expansion of $\langle W(K)\rangle/\langle W(\bigcirc)\rangle$ coming from (4).
But this could be done just for the few simple knots where $\rho(K)$ can be
calculated, so may be here regarded as a self-consistency check.
That’s the point where the algebraic properties of the gauge groups come out;
for the groups we are interested in, they are summarised in the following
table:

| dim $\mathcal{G}$ | dim $T$ | $c_{2}$ | $c_{v}$
---|---|---|---|---
SO(N) | $N(N-1)/2$ | $N$ | $(N-1)/4$ | $(N-2)/2$
Sp(N) | $N(N+1)/2$ | $N$ | $(N+1)/4$ | $(N+2)/2$
SU(N) | $N^{2}-1$ | $N$ | $(N^{2}-1)/2N$ | $N$

hence, from (4.2), we get respectively for SO(N) and Sp(N) the following
values for $a$

$\displaystyle a_{SO(N)}$ $\displaystyle=$ $\displaystyle
1-i\left(\frac{2\pi}{k}\right)\frac{N-1}{4}-\frac{1}{2}\left(\frac{2\pi}{k}\right)^{2}\left(\frac{N-1}{4}\right)^{2}+O\left(\frac{1}{k^{3}}\right)$
(4.4) $\displaystyle a_{Sp(N)}$ $\displaystyle=$ $\displaystyle
1-i\left(\frac{2\pi}{k}\right)\frac{N+1}{4}-\frac{1}{2}\Big{(}\frac{2\pi}{k}\Big{)}^{2}\left(\frac{N+1}{4}\right)^{2}+O\left(\frac{1}{k^{3}}\right)$

while for both orthogonal and symplectic groups the value found for z is:

$z=-\frac{i\pi}{k}+O\Big{(}\frac{1}{k^{3}}\Big{)}$ (4.5)

These results are consistent with the ones found in the previous section by
means of the variational method both for SO(N) and Sp(2). Moreover (4.4) and
(4.5) extend the series expansion in $2\pi/k$ up the second order. The fact
that $z$ has not the quadratic contribution could be guessed from the very
beginning because of the peculiar property of the Chern-Simons Lagrangian: the
inversion symmetry. This implies that a change in the sign of the coupling
constant $k$ is compensated by the inversion of the orientating of the
manifold. When a knot $K$ is embedded in $\mathcal{M}^{3}$ the change of
orientation of the manifold corresponds to a $\pi$ rotation or its mirror
image $\tilde{K}$, so $\langle W(K)\rangle\big{|}_{k}=\langle
W(\tilde{K})\rangle\big{|}_{-k}$. On the other hand from skein relations (2)
is easy to see that $D_{K}(a,z)=D_{\tilde{K}}(a^{-1},-z)$; combining it with
the inversion symmetry one gets some restriction on the k-functional
dependence of the variables $a$ and $z$:

$a(k)=a^{-1}(-k)\qquad z(k)=-z(-k)$ (4.6)

So even powers of $k$ were not expected in the $z$ expansion; as one can see
(4.4) and (4.5) fulfil the constraints (4.6). The easiest functions that are
compatible with the series expansions (4.4)-(4.5), their restrictions (4.6)
and the samples (4.3) are:

$a=\textrm{exp}\left[-i\frac{2\pi}{k}c_{2}(T)\right]\qquad z=-2\ i\
\textrm{sin}\left(\frac{\pi}{2k}\right)$

Furthermore observe that in the groups table there is a value of N for whom
two lines match: for $N=2$ all the values for Sp(2) and SU(2) coincide. So the
expectation value of a Wilson loop along a generic knot K agrees in both
cases. This special point is the one where the HOMFLY and Kauffman polynomials
overlap to give the Jones polynomial. This is exactly the same result we have
found with the variational approach in section 3.2, but now extended to the
second order. Another interesting feature that can be read from the table is
the analogy between the quantities of SO(-N) and Sp(N), in particular one can
note in (4) as Wilson loop expectation values of a SO(-N)-Chern-Simons theory
for a knot $K$ correspond to the ones of its mirror image $\tilde{K}$ for a
Sp(N)-CS theory:

$\langle W(K)\rangle\Big{|}_{SO(-N)}=(-1)^{m}\ \langle
W(\tilde{K})\rangle\Big{|}_{Sp(N)}$ (4.7)

For odd-multicomponent knots-links the correspondence hold up to a global
sign, where m is the number of components. The mirror image $\tilde{K}$ is
needed in order to have opposite the chirality in framing that compensate a
sign in the odd terms expansion. In terms of Dubrovnik polynomial (4.7) became
$D_{K}|_{SO(-N)}=D_{\tilde{K}}|_{Sp(N)}$, at least for proper knots. So again
what suggested by the variational approach can be coherently recovered and
extended by the perturbative one.
The ambient isotopic Dubrovnik-Kauffman polynomial is obtained, as usual, from
the regular one thanks to a writhe normalisation: $a^{-w(K)}D_{K}$.
Another remarkable feature of the variational and perturbative approaches is
that allow us to generalise at once the present treatment also to the non-
compact groups such as SO(m,n), which are the more interesting ones for
describe general relativity in 2+1 dimensions by the Chern-Simons theory.
Although from a classical point of view locally isomorphic groups represent
the same gauge theory, we have seen as at the quantum level expectation values
even of simple knots differ. Thus in case one wants to take advance of the
Chern-Simons formalism to study quantum properties of gravity he will have to
consider the issue of which is the ”good” group election. Actually the values
of the fundamental quantities as the Casimirs $c_{2},c_{v}$, the group’s
dimension dim$\mathcal{G}$ and the fundamental representation dimension
dim($T$) of SO(m,n) are not different from the SO(N) ones, whenever $m+n=N$.
Hence the topological quantity $\langle W(K)\rangle$ (4) is not affected by
the signature change of the Cartan-Killing metric121212Of course a gauge
description of gravity needs a further step: also a signature’s change in the
space-time coordinates, this is more problematic because all the treatment
done in this paper is for compact manifolds $\mathcal{M}^{3}$.. Up the author
knowledge invariant knot polynomials for SO(m,n) groups are not found by means
of any other methods; could be interesting to verify it with the help of more
rigorous mathematical tools such as quantum groups. Moreover the SO(m,n)
Chern-Simons theory got a richer structure than the SU(N) one. In fact others
non-equivalent Chern-Simons Lagrangian can be built from their Chern’s
characteristic classes apart from the Pontryagin; for instance is possible to
use also the Euler or Nieh-Yan topological invariants (see [23] for a review).
The expectation values of knotted Wilson loops weighted by this Chern-Simons
density remains a topological invariant, but possibly of different kind.

##  5 Correspondence with the Potts Model

In this section we try to build a bridge between the previous results about
first order expectation values of quantum holonomies along a knotted path and
some statistical system such as the Potts Model. Of course it is clear that an
exact equality can not hold since the Chern-Simons observables are knot
invariants while the Potts partition functions are not. Nevertheless something
can be said, but at the price of renouncing to the knot topological
invariance. First let us remind some fundamental facts about the Potts model
that we will be used afterwords.
It is found in [19] that the partition function of the Q-Potts Model of a
statistical lattice represented by a graph G is the _Potts state bracket_
$\\{K(G)\\}$ of the knot-link K dual to the graph G. That’s because this state
bracket expansion coincides exactly with the dichromatic polynomial, or the
Tutte polynomial, of the graph G. We remember the definition of the Potts
state bracket:

$\displaystyle i)$
$\displaystyle\qquad\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\\}=Q^{-1/2}v\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\\}+\\{\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\\}$
(5.1) $\displaystyle ii)$ $\displaystyle\qquad\\{\bigcirc\ K\\}=Q^{1/2}\
\\{K\\}$ $\displaystyle iii)$ $\displaystyle\qquad\\{\bigcirc\\}=Q^{1/2}$

To be more precise for any alternating knot or link K it is possible to
construct a graph lattice G(K) checkerboard shading its planar diagram and
assigning to each shadow a vertex and for each crossing a bound, as shown in
figure 7. Vice-versa for any two dimensional graph G one can associate its
dual knot K(G). Note that this is a one-to-one131313When the white region is
left outside. mapping between planar graphs and alternate knots and note that
any knot got its alternate representative, that is can be drawn as an
alternate planar diagram.

Figure 7: K(G) $\longleftrightarrow$ shading of K(G) $\longleftrightarrow$
emerging of lattice graph G inside K $\longleftrightarrow$ G(K)

Thus the Q-Potts partition function for a certain statistical lattice
$P_{G}(Q,t)$ is given by the dichromatic polynomial $Z_{G}(Q,v)$ of its graph
G (whenever $v=e^{J/kt}-1$) or by the Potts state bracket of its associated
knot $\\{K\\}$ as follows:

$P_{G(K)}(Q,t)\ =\
\sum_{\sigma}e^{\frac{J}{k_{B}t}\sum_{<i,j>}\delta(\sigma_{i},\sigma_{j})}\ =\
Q^{V/2}\\{K\\}(Q,v=e^{J/k_{B}t}-1)\ ,$ (5.2)

where $V$ is the number of vertex of the graph (i.e. the number of the
lattice’s sites or rather the number of shaded region of the knot), $t$ is the
temperature, $k_{B}$ the Boltzmann’s constant, $\sigma_{n}$ is one of the Q
possible states of the nth vertex and $J=\pm 1$ according to the ferromagnetic
or anti-ferromagnetic case.

###  5.1 SO(-2) & Sp(2) Holonomies and Q=4 Potts Model

First we consider a special case, that is when the Kauffman polynomial reduces
to the Kauffman state bracket $[K](q)$ (or to the Jones Polynomial whether
writhe normalised), which occurs for the SO(-2), Sp(2) 141414Correlated by
(4.7) or SU(2) Chern-Simons theory, as we have seen in section 3.2 and 4:

$\langle W(K)\rangle(z=q-q^{-1},a=-q^{3})=[K](q)\ .$

Then we perform a shift in the q-variable: $[K]\rightsquigarrow q^{c(K)}[K]$,
where $c(K)$ is the number of crossing in the knot K diagram. This shift is
the point where regular isotopical invariance of the Kauffman polynomial is
broken. So focusing just on the first order approximation, one gets the
following bracket $q^{c(K)}[K](q)\big{|}_{1^{st}-order}:=\ \ll
K\gg(1-i\pi/2k)$:

$\displaystyle i)$
$\displaystyle\quad\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\gg\
=q^{2}\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\gg+\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\gg\
=\Big{[}1-\frac{i\pi}{k}+O\Big{(}\frac{1}{k^{2}}\Big{)}\Big{]}\
\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\gg+\ll\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\gg$
(5.3) $\displaystyle ii)$ $\displaystyle\quad\ll\bigcirc\ K\gg\
=N+O\Big{(}\frac{1}{k^{2}}\Big{)}\ \ll K\gg$ $\displaystyle iii)$
$\displaystyle\quad\ll\bigcirc\gg\ =N+O\Big{(}\frac{1}{k^{2}}\Big{)}$

The analogy with the Potts state bracket (5.1) is now evident:

$\\{K\\}(Q,v)=\ \ll K\gg(\pm v^{1/2}Q^{-1/4}).$ (5.4)

Let now concentrate on the SO(-2) case, such that once the q-shift is
reabsorbed one recovers knot invariance, so $Q=N^{2}=4$. Using (5.2) and (5.4)
it is easy to see that $-2^{V}\ll K\gg$ represents the Q=4 Potts partition
function for the lattice graph associated to the knot K. In terms of the first
order Wilson loops expansion it reads:

$P_{G(K)}=Q^{V/2}\\{K\\}=N^{V}\ q^{c(K)}\ \langle
W(K)\rangle\Big{|}_{1^{st}-\mathrm{order}}$ (5.5)

An example may get things clearer: consider a 2x2 lattice graph G of figure 7
and its dual knot-link $K(G)$ (with $V=4$). From skein relations (5.1) (or
equally from the deletion-contraction rule that define the dichromatic
polynomial $Z_{G}(4,v)$) one gets the Q=4 Potts partition function for the
graph $G(K)$:

$Z_{G}(4,v)=4^{V/2}\\{K\\}=4^{2}(4^{2}+4\cdot 4v+6v^{2}+4\cdot
4^{-1}v^{3}+4^{-1}v^{4})$ (5.6)

while from the skein relations (5.3) one get the expectation value of the
holonomy along the knot $K(G)$, up to $O(1/k^{2})$:

$-2^{V}\ q^{c(K)}\ \langle
W(K)\rangle\Big{|}_{1^{st}-\mathrm{ord}}=2^{4}\big{(}1-\frac{i2\pi}{k}\big{)}\Big{[}16\big{(}1+\frac{i2\pi}{k}\big{)}-32\big{(}1+\frac{i\pi}{k}\big{)}+24-8\big{(}1-\frac{i\pi}{k}\big{)}+4\big{(}1-\frac{i2\pi}{k}\big{)}\Big{]}$

It’s easy to see that (5.5) is fullfilled imposing $v=-2+i2\pi/k$ in (5.6). So
the first order expectation value of the Wilson loop along a knotted path K
for a SO(-2)/Sp(2) Chern Simons theory can be extracted from the partition
function of a Q=4 Potts model of a lattice graph $G(K)$ dual to the knot $K$,
and vice-versa. This correspondence works well for any two dimensional lattice
graph, not just for regular ones like the sample presented in figure 7.
Even thought $\langle W(K)\rangle\big{|}_{1^{st}-\mathrm{order}}$ and
$P_{G}(K)$ are not exactly the same they share some features, for instance
their zeroes. So $\langle W(K)\rangle\big{|}_{1^{st}}$’s zeros can be
interpreted as the Fisher zeros of the statistical lattice associated to $K$,
which encode many important physical properties of the system. Also the
critical temperature $t_{c}$ (when the statistical system acquires conformal
invariance) of the Potts model can be easily read: In the knot formalism it
occurs where $\langle W(K)\rangle=\langle W(\tilde{K})\rangle$, that is when
$1-i\pi/k=1$, so in the limit $k\rightarrow\infty$, which means
$t_{c}=\frac{J}{k_{B}}\frac{1}{\textrm{ln}(\sqrt{Q}+1)}$.
It’s worth remark at this point that the SO(-2)/Sp(2) group (or even SU(2))
gives rise to the Jones polynomial too. This polynomial (at the non-
perturbative level) is known to describe the partition function of a
particular kind of Potts model with two Boltzmann factor, which is of
different kind respect to the standard Potts model considered here (see [14]
and [20]).
The correspondence holds also at the following orders of the perturbative
expansion, basically in the same way it works at the first order. For instance
one can obtain $\langle W(K)\rangle\big{|}_{2^{sd}-order}$ from the Q=4 Potts
partition function identifying $v$ and $Q$ as follows:

$\displaystyle v$ $\displaystyle\leftrightsquigarrow$
$\displaystyle-2\left[1-\frac{i\pi}{k}-\left(\frac{\pi}{k}\right)^{2}+O\left(\frac{1}{k^{3}}\right)\right]$
$\displaystyle Q^{\frac{1}{2}}$ $\displaystyle\leftrightsquigarrow$
$\displaystyle-2\left[1-\frac{1}{2}\left(\frac{\pi}{k}\right)^{2}+O\left(\frac{1}{k^{3}}\right)\right]$

The simple relation between $Q$ and $N$ is now spoiled and moreover this fact
makes the analogy between the two models purely formal because choosing a
particular Q imply fixing at the same time the temperature to a constant
value.

###  5.2 Sp(N) holonomies and Q-Potts Model

We would like to do something similar to previous subsection, but for generic
$N$. Now that procedure is less direct because the Kauffman polynomial can not
be cast in a simple form such as the state bracket [K]. To connect the two
theories, in particular to give the Q-Potts partition function a similar
structure to the Dubrovnik polynomial one, we can introduce a new bracket
polynomial $\|K\|(Q,v)$ defined by the following skein relations:

$\displaystyle i)$
$\displaystyle\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}}\|-\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{B.eps}}\|\
=(Q^{-1/4}v^{1/2}-Q^{1/4}v^{-1/2})\big{[}\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\|-\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\|\big{]}$
(5.7) $\displaystyle ii)$
$\displaystyle\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Rcurl.eps}}\|=(Q^{1/4}v^{1/2}+Q^{1/4}v^{-1/2})\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}}\|\
\
,\quad\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Lcurl.eps}}\|=(Q^{-1/4}v^{1/2}+Q^{3/4}v^{-1/2})\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}}\|$
$\displaystyle iii)$ $\displaystyle\quad\|\bigcirc\|=Q^{1/2}$ $\displaystyle
iv)$
$\displaystyle\quad\|\raisebox{-0.31pt}{\includegraphics[width=8.5359pt]{2cross+.eps}}\|=(Q^{-1/2}v+Q^{1/2}+Q^{1/2}v^{-1})\
\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}}\|+\
\|\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}}\|$

The Q-Potts partition function, in character of the dichromatic polynomial
$Z_{G(K)}(Q,v)$, has the following form in term of $\|K\|$:

$Z_{G}(Q,T)=Q^{V/2}[Q^{-1/4}v^{1/2}]^{c(K)}\|K\|.$

Even in this form $\|K\|$ is not a isotopical invariant of the knots, as
$\langle W(K)\rangle$ because the two coefficients in (5.7-$ii$) are not
reciprocal and (5.7-$iv$) does not satisfy the second Reidemeister move.
However there is a point where both (5.7-$ii,iv$) becomes invariant, that is
for $v=(-Q\pm\sqrt{Q^{2}-4Q})/2$. This value of the temperature is exactly the
one that relates the Potts model to the Khovanov homology [21]. Comparing the
$\|K\|(Q,v)$ bracket with the first order expectation value of the holonomy
$\langle W(K)\rangle\big{|}_{1^{st-ord}}$ one has to impose $Q=N^{2}$ and
$v=N(1-i\pi/k)$. So the $\|K\|(N,k)$ invariance occurs, in terms of the Chern-
Simons coupling constant $k$ and the fundamental representation dimension $N$,
just for $N=-2$, i.e the previous case we analysed in section 5.1.
Therefore for a generic $Q=N^{2}\neq 4$ is not possible to pass from the Potts
partition function to the first order Wilson loop expectation value as we did
for the $SO(N)/Sp(2)$ case. What can be done at most is define a generic
bracket polynomial which include both $P_{G}$ and $\langle W(K)\rangle$ and
specialises to one or the another for some values of its variables. This is
done in appendix A.

##  6 Comments and Conclusions

In this paper is analysed the relation between expectation values of Wilson
loop in three-dimensional SO(N) Chern-Simons field theory and an isotopic
invariant of knots, the Kauffman polynomial. This equivalence is achieved in a
simple intuitive knot variational approach borrowed by [3]’s and [5]’s scheme
which was elaborated for obtaining the Witten result: HOMFLY polynomial from
the SU(N) gauge group. The key point of this construction is based on the
existence of a Fierz identity for the infinitesimal generators of the group in
certain representations. With precisely the same interpretation of the
expectation value’s path variations and no other extra assumptions respect to
the original work, here we exactly get the conformal field theory known result
for SO(N): Kauffman polynomial. It suggests that the easy variational knot
approach, expressly built for SU(N), works well also for different gauge group
theories as SO(N). So its heuristic geometrical assumptions are endorsed.
Convinced of all that and encouraged by negative dimension group theory
suggestion we explored also the Sp(N) group getting the exact skein relation.
In particular in the simple Sp(2) case we are able to find its isotopic
invariant: the original Jones Polynomial.
Furthermore to enforce and extend those results, an independent procedure has
been performed, the quantum field theory method can not only full recover the
variational approach but also: improve its outcomes precision of an order of
magnitude, extend to groups with semi-definite Cartan-Killing metric as well
Sp(N) with $N\neq 2$ and most of all prove, up to $O(1/k^{3})$, the
correspondence between isotopy invariant polynomials from SO(N) and Sp(-N)
Chern-Simons theories.
To sum up, these procedures give for SU(N), SO(N)/Sp(N) and Sp(2) the famous
HOMFLY, Kauffman and Jones polynomials respectively. Hence they may be used
for other groups or representations to find new link invariants, both based on
skein relations or not. This could give new insights into knots theory, which
is still looking for a link invariant able to distinguish conclusively knots
isotopic equivalence.
From a physical point of view it’s interesting to note that not only the Jones
polynomial, at non perturbative level, correspond to the partition function of
the Potts model with two Boltzmann weight factors, but also its first order
perturbation expansion, in the realm of the Chern-Simons theory, gives the
standard Q=4 Potts partition function (and vice-versa). Moreover the
connection between the quantum holonomies of Sp(2) Chern-Simons theory and the
$Q=4$ Potts partition function opens the possibility to relate apparently
disconnected physical systems. This is actually the main motivation of the
author. In fact, since [22], it is well known that Sp(2)$\times$Sp(2) Chern-
Simons theory describes 2+1 gravity with negative cosmological constant.
Furthermore the first terms in the Kauffman bracket expansion give states of
3+1 quantum gravity in the loop representation [6]. This feature of knot
theory may represent the tip of an iceberg that links discrete statistical
models with the expectation value of holonomies of gravitational theories.
Work in this direction is in progress.

## Acknowledgements

I would like to thank Louis Kauffman, Roberto Troncoso, Steven Willison and
Jorge Zanelli for fruitful discussions.
The Centro de Estudios Científicos (CECS) is funded by the Chilean Government
through the Millennium Science Initiative and the Centers of Excellence Base
Financing Program of Conicyt and by Conicyt grant ”Southern Theoretical
Physics Laboratory” ACT-91. CECS is also supported by a group of private
companies which at present includes Antofagasta Minerals, Arauco, Empresas
CMPC, Indura, Naviera Ultragas and Telefónica del Sur.

##  Appendix A General Potts-Dubrovnik polynomial $M_{K}$

Define the following bracket polynomial $M_{K}(a,b,c,d,z)$:

$\displaystyle i)$ $\displaystyle\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{A.eps}})-M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{B.eps}})\
=z\big{[}M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}})-M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}})\big{]}$
$\displaystyle ii)$ $\displaystyle\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Rcurl.eps}})=a\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}})\quad,\quad
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Lcurl.eps}})=b\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{Arc.eps}})$
$\displaystyle iii)$ $\displaystyle\quad M(\bigcirc)=d$ $\displaystyle iv)$
$\displaystyle\quad
M(\raisebox{-0.31pt}{\includegraphics[width=8.5359pt]{2cross+.eps}})=c\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{C.eps}})+\
M(\raisebox{-0.25pt}{\includegraphics[width=14.22636pt]{D.eps}})$

$M_{K}$ reduces to the Kauffman-Dubrovnik polynomial when $b=a^{-1},\ c=0,\
d=1$; while to $\langle W(K)\rangle$ when $d=(a-a^{-1})/z+1$. So for those
values of the variables it is an invariant of regular isotopy. But the Potts
partition function is not invariant so this latter has $b\neq a^{-1}$ and $c$
switched on, as can see in the following table, where two different
specialisations of the $M_{K}$ polynomial are shown:

$M_{K}$ | $a$ | $b$ | $c$ | $d$ | $z$
---|---|---|---|---|---
$\langle W(K)\rangle$ | $\alpha$ | $\alpha^{-1}$ | $0$ | $(a-a^{-1})/z+1$ | $-i\pi/k$
$\|K(G)\|$ | $Q^{\frac{1}{4}}v^{\frac{1}{2}}+Q^{\frac{1}{4}}v^{\frac{-1}{2}}$ | $Q^{\frac{-1}{4}}v^{\frac{1}{2}}+Q^{\frac{3}{4}}v^{\frac{-1}{2}}$ | $Q^{-\frac{1}{2}}v+Q^{\frac{1}{2}}+Q^{\frac{1}{2}}v^{-1}$ | $Q^{\frac{1}{2}}$ | $Q^{-\frac{1}{4}}v^{\frac{1}{2}}-Q^{\frac{1}{4}}v^{-\frac{1}{2}}$

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Let $P$ be a set of points in a metric space $(X,\mathrm{dist})$. Let
$k,z\in\mathbb{N}$ and $\varepsilon\geqslant 0$ be parameters. Let $P$ be
partitioned into disjoint subsets $P_{1},\cdots,P_{s}$, and let
$Z=\\{z_{1},\cdots,z_{s}\\}$ be a set of numbers such that $\mbox{{\sc
opt}}_{k,z_{i}}(P_{i})\leqslant\mbox{{\sc opt}}_{k,z}(P)$ for each $P_{i}$. If
$P_{i}^{*}$ is an $(\varepsilon,k,z_{i})$-mini-ball covering of $P_{i}$ for
each $1\leqslant i\leqslant s$, then $\cup_{i=1}^{s}P_{i}^{*}$ is an
$(\varepsilon,k,z)$-mini-ball covering of $P$.

######  Proof A.2.

First of all, observe that the weight of the point set $P$ is preserved by the
union of the mini-ball coverings. Indeed, $\sum_{p\in
P}w(p)=\sum_{i=1}^{s}\sum_{p\in P_{i}}w(p)=\sum_{i=1}^{s}\sum_{q\in
P^{*}_{i}}w(q)$.

Next, consider an arbitrary point $p\in P$, and $P_{i}$ be the subset
containing $p$. Then $p$ has a representative point $q$ in the
$(\varepsilon,k,z_{i})$-mini-ball covering $P_{i}^{*}$ of $P_{i}$. By
Definition 2.1, $\mathrm{dist}(p,q)\leqslant\varepsilon\cdot\mbox{{\sc
opt}}_{k,z_{i}}(P_{i})\leqslant\varepsilon\cdot\mbox{{\sc opt}}_{k,z}(P)$,
which proves that $\cup_{i=1}^{s}P_{i}^{*}$ is an $(\varepsilon,k,z)$-mini-
ball covering of $P$.

### Proof of Lemma 2.4

######  Lemma 2.4 (Transitive Property).

Let $P$ be a set of $n$ points in a metric space $(X,\mathrm{dist})$. Let
$k,z\in\mathbb{N}$ and $\varepsilon,\gamma\geqslant 0$ be four parameters. Let
$P^{*}$ be a $(\gamma,k,z)$-mini-ball covering of $P$, and let $Q^{*}$ be an
$(\varepsilon,k,z)$-mini-ball covering of $P^{*}$. Then, $Q^{*}$ is an
$(\varepsilon+\gamma+\varepsilon\gamma,k,z)$-mini-ball covering of $P$.

######  Proof A.3.

Note that the weight-preservation property of mini-ball covering implies that
$\sum_{p\in P}w(p)=\sum_{p^{*}\in P^{*}}w(p^{*})=\sum_{q^{*}\in
Q^{*}}w(q^{*})$. It remains to show that any point $p\in P$ has a
representative point $q^{*}\in Q^{*}$ so that
$\mathrm{dist}(p,q^{*})\leqslant(\varepsilon+\gamma+\varepsilon\gamma)\cdot\mbox{{\sc
opt}}_{k,z}(P)$.

Since $P^{*}$ is an $(\gamma,k,z)$-mini-ball covering for $P$, there is a
representative point $p^{*}\in P^{*}$ for $p$ for which
$\mathrm{dist}(p,p^{*})\leqslant\gamma\cdot\mbox{{\sc opt}}_{k,z}(P)$.
Similarly, since $Q^{*}$ is an $(\varepsilon,k,z)$-mini-ball covering for
$P^{*}$, there is a representative point $q^{*}\in Q^{*}$ for $p^{*}$ such
that $\mathrm{dist}(p^{*},q^{*})\leqslant\varepsilon\cdot\mbox{{\sc
opt}}_{k,z}(P^{*})$. Hence,

$\begin{array}[]{lll}\mathrm{dist}(p,q^{*})&\leqslant&\mathrm{dist}(p,p^{*})+\mathrm{dist}(p^{*},q^{*})\\\
&\leqslant&\gamma\cdot\mbox{{\sc opt}}_{k,z}(P)+\varepsilon\cdot\mbox{{\sc
opt}}_{k,z}(P^{*})\\\ &\leqslant&\gamma\cdot\mbox{{\sc
opt}}_{k,z}(P)+\varepsilon\cdot(1+\gamma)\cdot\mbox{{\sc opt}}_{k,z}(P)\hskip
28.45274pt\mbox{ (by Definition\leavevmode\nobreak\ \ref{def:coreset})}\\\
&=&(\varepsilon+\gamma+\varepsilon\gamma)\cdot\mbox{{\sc
opt}}_{k,z}(P)\enspace.\end{array}$

We conclude that $q^{*}$ is a valid representative for $p$, thus finishing the
proof.

### Proof of Lemma 2.5

######  Lemma 2.5.

Let $P$ be a finite set of points in a metric space $(X,\mathrm{dist})$ of
doubling dimension $d$. Let $0<\delta\leqslant\mbox{{\sc opt}}_{k,z}(P)$, and
let $Q\subseteq P$ be a subset of $P$ such that for any two distinct points
$q_{1},q_{2}\in Q$, $\mathrm{dist}(q_{1},q_{2})>\delta$. Then $|Q|\leqslant
k\left(\frac{4\cdot\mbox{{\sc opt}}_{k,z}(P)}{\delta}\right)^{d}+z$.

######  Proof A.4.

Consider an optimal solution for the $k$-center problem with $z$ outliers on
$P$. Let $\mathcal{B}_{\mathrm{opt}}$ be the set of $k$ balls in this optimal
solution. Since $(X,\mathrm{dist})$ is a metric space of doubling dimension
$d$, every ball in $\mathcal{B}_{\mathrm{opt}}$ can be covered by at most
$(2\cdot\frac{\mbox{{\sc
opt}}_{k,z}(P)}{\delta/2})^{d}=(4\cdot\frac{\mbox{{\sc
opt}}_{k,z}(P)}{\delta})^{d}$ mini-balls of radius $\delta/2$. Besides, as the
distance between points of $Q$ is more than $\delta$, each mini-ball of radius
$\delta/2$ contains at most one point of $Q$. Therefore, the number of points
in $Q$ covered by $\mathcal{B}_{\mathrm{opt}}$ is at most
$(4\cdot\frac{\mbox{{\sc opt}}_{k,z}(P)}{\delta})^{d}$. Besides, as
$\mathcal{B}_{\mathrm{opt}}$ is an optimal solution, at most $z$ points of $Q$
are not covered by $\mathcal{B}_{\mathrm{opt}}$. Thus, $|Q|\leqslant
k(4\cdot\frac{\mbox{{\sc opt}}_{k,z}(P)}{\delta})^{d}+z$.

### Proof of Lemma 4.6

######  Lemma 4.6.

After the point $p_{t}$ arriving at time $t$ has been handled, we have: for
each point $p\in P(t)$ there is a representative point $q\in P^{*}$ such that
$\mathrm{dist}(p,q)\leqslant\varepsilon\cdot r$.

######  Proof A.5.

We may assume by induction that the lemma holds after the previous point has
been handled. (Note that the lemma trivially holds before the arrival of the
first point.) We now show that the lemma also holds after processing $p_{t}$.

It is easily checked that after lines 4 of the algorithm, there is indeed a
representative in $P^{*}$ within distance $\varepsilon\cdot r$, namely the
point $q$ in line 2 or $p$ itself in line 4.

In each iteration of the while-loop in lines 8, the value of $r$ is doubled
and UpdateCoreset is called. We will show that the lemma remains true after
each iteration. Let $r^{-}$ and $r^{+}$ denote the value of $r$ just before
and after updating it in line 9, respectively, so $r^{+}=2\cdot r^{-}$. Let
$p$ be an arbitrary point in $P(t)$. Let $q^{-}$ be the representative point
of $p$ before the call to UpdateCoreset. Because the statement of the lemma
holds before the call, we have
$\mathrm{dist}(p,q^{-})\leqslant\varepsilon\cdot r^{-}$. Let $q^{+}$ denote
the representative point of $q^{-}$ just after the call to UpdateCoreset.
(Possibly $q^{+}=q^{-}$.) Since the distance parameter $\delta$ of
UpdateCoreset in the call is set to $(\varepsilon/2)\cdot r^{+}$, we know that
$\mathrm{dist}(q^{-},q^{+})\leqslant(\varepsilon/2)\cdot r^{+}$. Hence,

$\mathrm{dist}(p,q^{+})\leqslant\mathrm{dist}(p,q^{-})+\mathrm{dist}(q^{-},q^{+})\leqslant\varepsilon\cdot
r^{-}+\frac{\varepsilon}{2}\cdot
r^{+}\leqslant\varepsilon\cdot\frac{r^{+}}{2}+\frac{\varepsilon}{2}\cdot
r^{+}=\varepsilon\cdot r^{+}\enspace,$

which finishes the proof of the lemma.

### Proof of Lemma 7.1

######  Lemma 7.1 ([11]).

$\mathbf{Pr}\left[\forall_{1\leqslant i\leqslant m}|P_{i}\cap
P_{\mathrm{out}}|\leqslant\frac{6z}{m}+3\log{n}\right]\geqslant 1-1/n^{2}$.

######  Proof A.6.

Consider an optimal solution for the $k$-center problem with $z$ outliers on
$P$. Let $\mathcal{B}_{\mathrm{opt}}$ be the set of $k$ balls in this optimal
solution and let $P_{\mathrm{out}}=\\{q_{1},\cdots,q_{z}\\}\subset P$ be the
outliers, that is, the points not covered by the balls in
$\mathcal{B}_{\mathrm{opt}}$. Let us consider a random variable $X_{i}$ that
corresponds to the number of outliers that are assigned to an machine $M_{i}$
for $i\in[m]$. First of all, observe that $\mathbf{E}[X_{i}]=\frac{z}{m}$.
Next, we consider $X_{i}=\sum_{j=1}^{z}Y_{ij}$ where every random variable
$Y_{ij}$ is an indicator random variable which is one of the outlier $q_{j}$
is assigned to the machine $M_{i}$ and zero otherwise. Now, we use the
Chernoff bound to show that $X_{i}$ is concentrated around its expectation.

######  Lemma A.7 (Multiplicative Chernoff bound).

[15, 17] Let $X_{1},\cdots,X_{N}$ be independent random variables, with
$\mathbf{Pr}\left[X_{i}=1\right]=p$ and $\mathbf{Pr}\left[X_{i}=0\right]=1-p$
for each $i$ and for certain $0\leqslant p\leqslant 1$. Let
$X=\sum_{i=1}^{N}X_{i}$. Then,

  * •

for any $\tau\geqslant 6\cdot\mathbf{E}[X]$, we have
$\mathbf{Pr}\left[X\geqslant\tau\right]\leqslant 2^{-\tau}\enspace.$

We let $\tau=\frac{6z}{m}+3\log{n}$, then, $\tau\geqslant
6\cdot\mathbf{E}[X_{i}]$. Thus, using the Chernoff bound we have

$\mathbf{Pr}\left[X_{i}\geqslant\frac{6z}{m}+3\log{n}\right]=\mathbf{Pr}\left[X_{i}\geqslant\tau\right]\leqslant
2^{-\tau}\leqslant 2^{-(\frac{6z}{m}+3\log{n})}\leqslant
2^{-3\log{n}}=1/n^{3}\enspace.$

Now, since we have $m=\sqrt{n}$ machines, we use a union bound to obtain

$\mathbf{Pr}\left[\exists_{i\in[m]}X_{i}\geqslant\frac{6z}{m}+3\log{n}\right]\leqslant\sum_{i+1}^{m}\mathbf{Pr}\left[X_{i}\geqslant\frac{6z}{m}+3\log{n}\right]\leqslant
m/n^{3}\leqslant 1/n^{2}\enspace.$

##  Appendix B Omitted proofs of the $\Omega(k/\varepsilon^{d})$ lower bound
for the streaming model

This section provides the missing proofs for our lower bounds. Recall that
$P^{*}(t^{\prime})\subseteq P^{*}(t)\cup P^{-}\cup P^{+}$ is the coreset
maintained by the algorithm, which is supposed to be an
$(\varepsilon,k,z)$-coreset of the point set $P(t^{\prime})=P(t)\cup P^{-}\cup
P^{+}$.

######  Lemma B.1.

The optimal $k$-center radius with $z$ outliers of the
$(\varepsilon,k,z)$-coreset $P^{*}(t^{\prime})$ is at most $r$, i.e.,
$\mbox{{\sc opt}}_{k,z}(P^{*}(t^{\prime}))\leqslant r$.

######  Proof B.2.

To this end, we show that there exist $k$ balls centered at $k$ centers of
radius $r$ that can cover all points in $(C_{1}\cup\ldots\cup C_{k-2d+1}\cup
P^{+}\cup P^{-})\setminus\\{p^{*}\\}$. Recall that
$p^{*}=(p_{1}^{*},\ldots,p_{d}^{*})$ is the point that is not explicitly
stored in $P^{*}(t)$ and we assume that $p^{*}$ belongs to a cluster
$C_{i^{*}}$ for $i^{*}\in[k-2d+1]$.

Observe that for every $i\neq i^{*}$, since the diameter of $C_{i}$ is
$\sqrt{d}\lambda$, all points of $C_{i}$ can be covered by a ball of radius
$\sqrt{d}\lambda/2$. We assumed that $\lambda:=1/(4d\varepsilon)$ is an
integer, $h:=d(\lambda+2)/2$ and $r:=\sqrt{h^{2}-2h+d}$. As $d\geqslant 1$, we
observe that $\sqrt{d}\lambda/2\leqslant r$.

Figure 8: Illustration of the lower bound for the streaming model. Here,
$P^{*}(t^{\prime})$ underestimates $\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))$
since $2d$ balls of radius $r$ can cover $P^{+}\cup P^{-}\cup
C_{i^{*}}\setminus\\{p^{*}\\}$ (dashed balls), and then $\mbox{{\sc
opt}}_{k,z}(P^{*}(t^{\prime}))\leqslant r$. However, $\mbox{{\sc
opt}}_{k,z}(P(t^{\prime}))=(r+h)/2$ (the red ball).

To cover $C_{i^{*}}\cup P^{-}\cup P^{+}$, we define $2d$ centers
$c_{1}^{+},\ldots,c_{d}^{+}$ and $c_{1}^{-},\ldots,c_{d}^{-}$, where
$c_{j}^{+}=(c_{j,1}^{+},\ldots,c_{j,d}^{+})$ such that
$c_{j,j}^{+}:=p_{j}^{*}+h$ and $c_{j,\ell}^{+}:=p_{\ell}^{*}$ for all
$\ell\neq j$. Similarly, $c_{j}^{-}=(c_{j,1}^{-},\ldots,c_{j,d}^{-})$ such
that $c_{j,j}^{-}:=p_{j}^{*}-h$ and $c_{j,\ell}^{-}:=p^{*}_{\ell}$ for all
$\ell\neq j$; see Figure 8. We claim that $2d$ balls centered at these $2d$
centers of radius $r$ cover all points in $P^{+}\cup P^{-}\cup
C_{i^{*}}\setminus\\{p^{*}\\}$.

For the moment suppose this claim is correct. The number of clusters $C_{i}$
where $i\neq i^{*}$ is $k-2d$. We just showed that all points of $C_{i}$ can
be covered by a ball of radius $r$. We also claimed (which needs to be proved)
that there exist $2d$ balls centered at these $2d$ centers of radius $r$ cover
all points in $P^{+}\cup P^{-}\cup C_{i^{*}}\setminus\\{p^{*}\\}$. In
addition, in Claim 7 (below) we prove that the total weight of the outlier
points $o_{1},\ldots,o_{z}$ in $P^{*}(t^{\prime})$ is at most $z$. Thus,
$\mbox{{\sc opt}}_{k,z}(P^{*}(t^{\prime}))\leqslant r$ as we want to prove.

Next, we prove the claim. Indeed, for each $p_{j}^{+}\in P^{+}$, we have
$\mathrm{dist}(p_{j}^{+},c_{j}^{+})=r$. Similarly, for each $p_{j}^{-}\in
P^{-}$, we have $\mathrm{dist}(p_{j}^{-},c_{j}^{-})=r$. Let
$q=(q_{1},\ldots,q_{d})$ be an arbitrary point in
$C_{i^{*}}\setminus\\{p^{*}\\}$. We define
$j_{q}:=\arg\max_{\ell\in[d]}{|q_{\ell}-p^{*}_{\ell}|}$ to be the dimension
along which $p^{*}$ and $q$ have the maximum distance from each other, and let
$\mu_{q}:=|q_{j_{q}}-p_{j_{q}}^{*}|$ be the distance along the dimension
$j_{q}$. Observe that $\mu_{q}\geqslant 1$.

######  Claim 5.

For an arbitrary point $q=(q_{1},\ldots,q_{d})\in
C_{i^{*}}\setminus\\{p^{*}\\}$, we have the following bounds:

  * •

If $q_{j_{q}}-p_{j_{q}}^{*}>0$, then for the center
$c_{j_{q}}^{+}\in\\{c_{1}^{+},\ldots,c_{d}^{+}\\}$ we have
$\mathrm{dist}(q,c_{j_{q}}^{+})\leqslant r$.

  * •

If $q_{j_{q}}-p_{j_{q}}^{*}<0$, then for the center
$c_{j_{q}}^{-}\in\\{c_{1}^{-},\ldots,c_{d}^{-}\\}$ we have
$\mathrm{dist}(q,c_{j_{q}}^{-})\leqslant r$.

  * _Proof._ First assume that $q_{j_{q}}-p_{j_{q}}^{*}>0$. The other case is proven similarly. We let $c_{j_{q}}^{+}=(c_{j_{q},1}^{+},\ldots,c_{j_{q},d}^{+})$. Recall that $c_{j_{q},j_{q}}^{+}=p_{j_{q}}^{*}+h$, and $c_{j_{q},\ell}^{+}=p_{\ell}^{*}$ for all $\ell\neq j_{q}$. Therefore, $|q_{j_{q}}-c_{j_{q},j_{q}}^{+}|=|q_{j_{q}}-(p^{*}_{j_{q}}+h)|=|\mu_{q}-h|$ and for all $\ell\neq j_{q}$ we have $|q_{\ell}-c_{j_{q},\ell}^{+}|=|q_{\ell}-p^{*}_{\ell}|\leqslant\mu_{q}$. Thus,

$\mathrm{dist}(q,c_{j_{q}}^{+})=\sqrt{\sum_{\ell=1}^{d}|q_{\ell}-c_{j_{q},\ell}^{+}|^{2}}\leqslant\sqrt{(d-1)\mu_{q}^{2}+(\mu_{q}-h)^{2}}=\sqrt{h^{2}+d\mu_{q}^{2}-2\mu_{q}h}\enspace.$

Since $0<\mu_{q}\leqslant\lambda$ we have
$h=\frac{d}{2}(\lambda+2)\geqslant\frac{d}{2}(\mu_{q}+1)$. As
$\mu_{q}\geqslant 1$, we can multiply both sides of this inequality by
$2(\mu_{q}-1)$ to obtain
$h\cdot(2\mu_{q}-2)\geqslant\frac{d}{2}(\mu_{q}+1)\cdot
2(\mu_{q}-1)=d(\mu_{q}^{2}-1)$ what yields $-2h+d\geqslant
d\mu_{q}^{2}-2\mu_{q}h$. Finally, by adding $h^{2}$ to both sides we have
$h^{2}-2h+d\geqslant h^{2}+d\mu_{q}^{2}-2\mu_{q}h$. Recall that
$r=\sqrt{h^{2}-2h+d}$ and
$dist(q,c_{j_{q}}^{+})\leqslant\sqrt{h^{2}+d\mu_{q}^{2}-2\mu_{q}h}$. Thus
$dist(q,c_{j_{q}}^{+})\leqslant r$ that proves this claim. $\lhd$

Next, we prove that $\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))=(h+r)/2$.

######  Claim 6.

$\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))=(h+r)/2$.

  * _Proof._ We proved that $\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))\geqslant(h+r)/2$. Now, we prove that $\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))\leqslant(h+r)/2$. In fact, in Claim 5, we proved that the balls in $\\{b(c_{1}^{-},r),\ldots,b(c_{d}^{-},r)\\}\cup\\{b(c_{1}^{+},r),\ldots,b(c_{d}^{+},r)\\}$ cover all points in $P^{+}\cup P^{-}\cup C^{i^{*}}\setminus\\{p^{*}\\}$. We define a center $c^{\prime}=(c^{\prime}_{1},\ldots,c^{\prime}_{d})$ such that $c^{\prime}_{1}:=p_{1}^{*}-(h+r)/2$ and $c^{\prime}_{\ell}:=p^{*}_{\ell}$ for all $\ell\neq 1$. Note that $h\geqslant r$ and $c_{1}^{-}=(c_{1,1}^{-},\ldots,c_{1,d}^{-})$ such that $c_{1,1}^{-}:=p_{1}^{*}-h$ and $c_{j,\ell}^{+}:=p^{*}_{\ell}$ for all $\ell\neq 1$. Thus, $\mathrm{dist}(c_{1}^{-},c^{\prime})=|c_{1,1}^{-}-c^{\prime}_{1}|=|p_{1}^{*}-h-(p_{1}^{*}-(h+r)/2)|=|(h-r)/2|=(h-r)/2$. Now, we observe that using the triangle inequality, $b(c_{1}^{-},r)\subseteq b(c^{\prime},(h+r)/2)$. (In Figure 8, the ball $b(c^{\prime},(h+r)/2)$ is shown in red.) Indeed, let $q$ be an arbitrary point in $b(c_{1}^{-},r)$. Using the triangle inequality, we have $\mathrm{dist}(q,c^{\prime})\leqslant\mathrm{dist}(q,c_{1}^{-})+\mathrm{dist}(c_{1}^{-},c^{\prime})\leqslant r+(h-r)/2=(h+r)/2$, which means $q\in b(c^{\prime},(h+r)/2)$. Therefore, $b(c_{1}^{-},r)\subseteq b(c^{\prime},(h+r)/2)$. In addition, observe that the ball $b(c^{\prime},(h+r)/2)$ covers the point $p^{*}$. Now, if we replace the ball $b(c_{1}^{-},r)$ by $b(c^{\prime},(h+r)/2)$, the union of balls $\\{b(c^{\prime},(h+r)/2)\\}\cup\\{b(c_{2}^{-},r),\ldots,b(c_{d}^{-},r)\\}\cup\\{b(c_{1}^{+},r),\ldots,b(c_{d}^{+},r)\\}$ will cover $C_{i^{*}}\cup P^{+}\cup P^{-}$. This essentially means that $\mbox{{\sc opt}}_{k,z}(P+2d)\leqslant(h+r)/2$. $\lhd$

It remains to argue that the total weight of the outlier points
$o_{1},\ldots,o_{z}$ in $P^{*}(t^{\prime})$ is at most $z$.

######  Claim 7.

The total weight of the outlier points $o_{1},\ldots,o_{z}$ in
$P^{*}(t^{\prime})$ is at most $z$.

  * _Proof._ Suppose this is not the case. We consider an optimal set $\mathcal{B}^{*}$ of $k$ balls that covers the weighted points in $P^{*}(t^{\prime})$ except a total weight of at most $z$. Since $P^{*}(t^{\prime})$ is an $(\varepsilon,k,z)$-coreset for $P(t^{\prime})$, the radius of balls in the set $\mathcal{B}^{*}$ is at most $(1+\varepsilon)\cdot\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))=(1+\varepsilon)\cdot(h+r)/2$, where we use Claim 6 that shows $\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))=(h+r)/2$.

Suppose for the sake of contradiction, the total weight of the outlier points
$o_{1},\ldots,o_{z}$ in $P^{*}(t^{\prime})$ is more than $z$, thus, at least
one outlier point, say $o_{1}$, must be covered by a ball from
$\mathcal{B}^{*}$. On the other hand, the nearest non-outlier point to $o_{1}$
is at distance at least $4(h+r)$ from $o_{1}$. Thus, since the radius of balls
in the optimal set $\mathcal{B}^{*}$ is at most $(1+\varepsilon)\cdot(h+r)/2$,
such an outlier $o_{1}$ must be a singleton in its ball. Let
$k^{\prime}\geqslant 1$ be the number of outliers that are covered by
singleton balls from $\mathcal{B}^{*}$.

As $P^{*}(t^{\prime})$ is an $(\varepsilon,k,z)$-coreset of $P(t^{\prime})$,
if we expand the radius of the balls in $\mathcal{B}^{*}$ by
$\varepsilon\cdot\mbox{{\sc opt}}_{k,z}(P(t^{\prime}))=\varepsilon(h+r)/2$,
then the total weight of points in $P(t^{\prime})$ that are not covered by
these expanded balls is at most $z$. Therefore, the points in
$C_{1}\cup\ldots\cup C_{k-2d+1}\cup P^{+}\cup P^{-}$ need to be covered by the
remaining $k-k^{\prime}$ expanded balls.

Consider the following $k$ sets: $2d$ sets
$\\{p^{+}_{1}\\},\ldots,\\{p^{+}_{d}\\}$ and
$\\{p^{-}_{1}\\},\ldots,\\{p^{-}_{d}\\}$, as well as $k-2d$ sets $C_{i}$,
where $i\neq i^{*}$. Since the pairwise distances between these $2d+(k-2d)=k$
sets are at least $\sqrt{2}(h+r)$. Besides, the radius of the expanded balls
is at most $(1+2\varepsilon)(h+r)/2$, where $1+2\varepsilon<\sqrt{2}$ since we
assume $\varepsilon\leqslant\frac{1}{8d}$. Hence, each of the remaining
$k-k^{\prime}$ expanded balls can cover at most one of these $k$ sets. As
$k-k^{\prime}$ balls of $\mathcal{B}^{*}$ remained for these $k$ sets, then at
least $k^{\prime}$ sets cannot be covered by the expanded balls.

We assumed the weight of every point $p^{+}_{i}$ (or $p^{-}_{i}$) is two444
However, we can in fact change these weighted points to unweighted points by
replacing each weighted point $p^{+}_{i}$ (similarly $p^{-}_{i}$) by two
unweighted points $p^{+}_{i,0}$ and $p^{+}_{i,1}$, where $p^{+}_{i,0}$ is at
the same place of $p^{+}_{i}$, and $p^{+}_{i,1}$ is on the boundary of the
ball $b(c^{+}_{i},r)$ at distance $\varepsilon$ of $p^{+}_{i,0}$ ;see Figure
8. It is simple to see that our arguments still hold for this unweighted case
as
$\mathrm{dist}(p^{-}_{i,0},c^{+}_{i})=\mathrm{dist}(p^{-}_{i,1},c^{+}_{i})=r$,
and $\mathrm{dist}(p^{-}_{i,0},p^{+}_{i})=\varepsilon$.. In addition, the
number of points in every $C_{i}$ is at least $2$ since $\lambda\geqslant 2$
and $|C_{i}|=(\lambda+1)^{d}$. Recall that the union of balls in
$\mathcal{B}^{*}$ does not cover $z-k^{\prime}$ points in
$\\{o_{1},\ldots,o_{z}\\}$. Therefore, the total weight of the points that the
expanded balls do not cover is at least
$(z-k^{\prime})+2k^{\prime}=z+k^{\prime}$. As the total weight of outliers
must be at most $z$, we must have $k^{\prime}=0$, which is a contradiction to
the assumption that at least one outlier point must be covered by a ball from
$\mathcal{B}^{*}$. That is, the total weight of the outlier points
$o_{1},\ldots,o_{z}$ in $P^{*}(t^{\prime})$ is at most $z$ as we want. $\lhd$

######  Lemma B.3.

Let $0<\varepsilon\leqslant\frac{1}{8d}$. Let $\lambda:=1/(4d\varepsilon)$ be
an integer, $h:=d(\lambda+2)/2$ and $r:=\sqrt{h^{2}-2h+d}$. Then,
$r<(1-\varepsilon)(r+h)/2$.

######  Proof B.4.

We start with the statement that we want to prove, and then derive a sequence
of equivalent statements, until we arrive at a statement that is easily seen
to be true. Indeed, assume that $r<(1-\varepsilon)(r+h)/2$ is correct. Then,
we have $2r<r+h-\varepsilon r-\varepsilon h$ which means that
$r(1+\varepsilon)<h(1-\varepsilon)$. Since both sides of the inequality are
non-negative, we can raise them to the power of two to obtain
$r^{2}(1+\varepsilon)^{2}<h^{2}(1-\varepsilon)^{2}$. Since
$r=\sqrt{h^{2}-2h+d}$ and $h^{2}-2h+d\geqslant 0$, we have
$(h^{2}-2h+d)(1+2\varepsilon+\varepsilon^{2})<h^{2}(1-2\varepsilon+\varepsilon^{2})$
which means that
$h(-4h\varepsilon+1)+(h-d)+\varepsilon^{2}(2h-d)+2\varepsilon(2h-d)>0$.

We observe that all these four terms of this inequality are non-negative which
proves the claim. First of all, since $\varepsilon\leqslant\frac{1}{8d}$, we
observe that $-4h\varepsilon+1\geqslant 0$, and so, the first term is at least
zero. Next, as $\lambda\geqslant 1$, we have $h=d(\lambda+2)/2>d$ that implies
that $2h-d>h-d>0$. Hence, the second, third, and forth terms are greater than
zero. Thus the left-hand side of the above inequality is greater than zero and
this finishes the proof.