centertableaux,smalltableaux centertableaux 11institutetext: Department of
Physics, Indian Institute of Technology Bombay
Mumbai-400076, India  22institutetext: Indian Institute of Science Education
and Research Bhopal
Bhopal bypass, Bhopal 462066, India 33institutetext: Mandelstam Institute for
Theoretical Physics, School of Physics, University of the Witwatersrand,
Johannesburg, WITS 2050, South Africa

#  $U(N)$ Torus Link Invariants in the Large $N$ limit from Matrix Model
Approach

Archana Maji<EMAIL_ADDRESS>2,3 Kushal Chakraborty
<EMAIL_ADDRESS>2 Suvankar Dutta<EMAIL_ADDRESS>1 P. Ramadevi
<EMAIL_ADDRESS>
###### Abstract

In this paper we study $U(N)$ colored HOMFLY-PT polynomials of torus links in
the double scaling limit (polynomial variable $q\rightarrow 1$,
$N\rightarrow\infty$ keeping $q^{N}$ fixed). We show that, in this limit, the
colored HOMFLY-PT polynomial of any $(L\alpha,L\beta)$ torus link can be
expressed in terms of the colored HOMFLY-PT polynomial of $(L,L)$ torus link.
Using the connection between matrix models and the Chern-Simons field
theoretic invariants, we show that the colored torus link invariants are
uniquely expressed in terms of connected correlation functions of operators in
$U(N)$ matrix model. We determine the leading and subleading contribution to
some of the correlators at large $N$ from the matrix model approach and find
that they match exactly with those obtained from the corresponding colored
HOMFLY-PT polynomials.

##  1 Introduction

The famous discovery of Jones polynomial jones1985polynomial ; jones1987hecke
by Vaughan Jones in 1984 led to a rejuvenation of the mathematical theory of
knots and links. A knot $\mathcal{K}$ is an embedding of a circle $S^{1}$
inside a three-manifold $M$ whereas a link $\mathcal{L}$ is a collection of
two or more knots linked in a nontrivial way. Two knots $\mathcal{K}_{1}$ and
$\mathcal{K}_{2}$ are considered to be equivalent if one can be deformed to
another through any of the three Reidemeister moves. The central problem of
knot theory is to classify different knots and links. To address this
classification problem there comes various polynomial invariants viz Alexander
polynomial alexander1928topological , Jones polynomial, HOMFLY-PT polynomial
freyd1985new ; przytycki2016invariants  et cetra in the order of increasing
sophistication.

Knot theory has become of spectacular interest to physicists ever since
Witten’s seminal work giving an intrinsically three dimensional definition of
Jones polynomial from the perspective of quantum field theory
witten1989quantum . The Chern-Simons theory, a three dimensional gauge theory
with an underlying gauge group $G$ provides a natural framework to compute the
topological invariants of three-manifolds and of knots and links embedded
inside those three-manifolds. In particular, the correlation function of the
observables of Chern-Simons theory, viz. Wilson loop operators, is nothing but
Jones polynomial when the gauge group at consideration is $SU(2)$ and the
representations put on the Wilson loops are that of the fundamental
representation $\bigg{(}R\equiv\text{\ydiagram[]{1}}\bigg{)}$ of $SU(2)$.
Likewise, with the fundamental representations of $G=SU(N)$ and $SO(N)$ we
obtain HOMFLY-PT polynomial and Kauffman polynomial kauffman1990invariant
respectively. If other higher dimensional representations are placed on the
Wilson loops then we obtain the corresponding colored polynomials.

In this work we confine ourselves to a class of links called torus links that
can be embedded on the surface of a torus in such a way that they do not cross
over themselves. Any $L$ component torus link can be denoted as
$(L\alpha,L\beta)$ where $L\alpha$ and $L\beta$ denote the number of times the
link wraps around the meridian and the longitude of the torus respectively.
Here $\alpha$ and $\beta$ are coprime to each other. These torus links can be
obtained as a closure of $L\alpha$ strand braid with $(L\alpha-1)L\beta$
crossings. We will study these torus link invariants in the double scaling
limit of Chern-Simons theory associated with gauge group $U(N)$. This limit is
defined as follows

$\displaystyle N,k\rightarrow\infty~{}~{}~{}\text{such
that}~{}~{}\lambda=\frac{N}{N+k}=\textrm{fixed},$ (1)

where $k$ is the coupling constant of the Chern-Simons theory. In terms of the
variable $q=\exp\left(\frac{2\pi i}{k+N}\right)$ the above limit can be
rephrased as

$\displaystyle q\rightarrow 1,N\rightarrow\infty~{}~{}~{}\textrm{such
that}~{}q^{N}=\textrm{fixed}.$ (2)

By two of the authors, it has been shown that in this limit the invariants for
a specific set of torus links viz. $(2,2m)$ torus links can be expressed in
terms of $(2,2)$ torus link (also called Hopf link) invariant
chakraborty2022large . Interestingly, this result can be generalised for any
arbitrary $L$ component torus link. In particular, we show that in the double
scaling limit we can express the $(L\alpha,L\beta)$ torus link invariant in
terms of $(L,L)$ torus link invariant.

Although the partition function of Chern-Simons theory can be obtained for
some three-manifolds, the calculation of correlation functions is difficult in
general. However, in the double scaling limit one can use the saddle point
approximation to compute some of the correlation functions. In particular, the
correlation function that corresponds to $(2,2)$ torus link invariant was
computed in Ref. chakraborty2022large  by mapping the problem to that of an
incompressible fluid with intial and final fluid densities being related to
the representations placed on the two component knots. In this paper we
further study the correlation functions corresponding to $(L,L)$ torus link in
the double scaling limit, for any arbitrary $L$. We obtain an exact expression
for the leading contribution of such link invariants using the saddle point
approximation Chakraborty:2021oyq ; chakraborty2022large . We validate this
result by explicitly computing the colored HOMFLY-PT polynomials for $(3,3)$
torus link carrying same/different representations on its three components.

It is well known in knot theory, that all knot invariants reduce to unknot
invariant in the double scaling limit. Similarly any $L$ component link
invariant reduces to the product of $L$ unknot invariants. The intertwining
between the components is not captured in this limit. These observations are
also seen in the matrix model results using saddle point approximation.
Specifically, the leading order contribution to any $n$-point correlator in
the large $N$ limit is the product of $n$ one-point correlators. Hence, we
need to investigate the subleading corrections to capture the intertwining
between the component knots constituting the link. The subleading corrections
to the knot invariants are captured by the connected piece of correlation
functions in the matrix model approach. We use the techniques developed in
Ref. eynard2015random ; Ambjorn:1992gw  to compute the connected pieces. We
then validate the results for the subleading contributions of Hopf link
invariants for some specific representations placed on the components of the
link. Further, we show that _any_ knot or link invariant modulo the leading
contribution (i.e. subtracting the leading contribution from the link
invariant) can be uniquely written in terms of connected correlation function
of operators in $U(N)$ matrix model. These connected correlators are related
to the reformulated invariants (which are written in terms of the colored
HOMFLY-PT polynomials)ramadevi2012reformulated .

The organisation of the paper is as follows: In section 2 we review how to
obtain the correlation function of links in $U(N)$ Chern-Simons theory. In
section 3 we write down explicitly the $U(N)$ invariant of $(2,2m)$ torus link
embedded inside three manifold $S^{3}$. In subsection 4.1 we review the double
scaling limit of $(2,2m)$ torus link invariant chakraborty2022large . The next
subsection 4.2 contains the generalisation of this result for any $L$
component $(L\alpha,L\beta)$ torus link invariant. In section 5 we discuss how
one can express the $(L,L)$ torus link invariant in terms of correlation
function involving $L$ Schur polynomials. An analytic expression for the
leading contribution of any $(L,L)$ torus link invariant is obtained in the
large $N$ limit using the saddle point approximation. Particularly we validate
this result by tabulating the colored HOMFLY-PT polynomial of $(3,3)$ torus
link. Subleading contributions to the link invariants have been discussed in
section 6. In that same section we also argue how the torus knot/link
invariants with arbitrary representations can be expressed in terms of
connected correlation functions of operators in $U(N)$ matrix model. Finally,
in the concluding section 7 we summarise our results and elaborate on the
future outlook.

##  2 Link invariants in $U(N)$ Chern-Simons theory

The three dimensional Chern-Simons theory on a manifold $M$ corresponding to
the gauge group $U(N)$ can be written as a sum of $SU(N)$ and $U(1)$ Chern-
Simons actions borhade2004u :

$\displaystyle S$
$\displaystyle=\frac{k}{4\pi}\int_{M}\text{Tr}\bigg{(}A\wedge
dA+\frac{2}{3}A\wedge A\wedge
A\bigg{)}+\frac{k_{1}}{4\pi}\int_{M}\text{Tr}\bigg{(}B\wedge dB\bigg{)}$
$\displaystyle\equiv\frac{k}{4\pi}\int_{M}d^{3}x~{}\epsilon^{\mu\nu\rho}~{}\text{Tr}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right)+\frac{k_{1}}{4\pi}\int_{M}d^{3}x~{}\epsilon^{\mu\nu\rho}~{}\text{Tr}\left(B_{\mu}\partial_{\nu}B_{\rho}\right)$
(3)

where $A$, $B$ are the gauge connections for the gauge groups $SU(N)$, $U(1)$
respectively and $k$, $k_{1}$ are the respective coupling constants. Thus, the
partition function of $U(N)$ Chern-Simons theory obtained, by integrating over
gauge inequivalent connections, is a product of two partition functions

$\displaystyle Z^{U(N)}[M]$
$\displaystyle=\int_{M}[\mathcal{D}A][\mathcal{D}B]e^{iS}\equiv
Z^{SU(N)}[M]\times Z^{U(1)}[M].$ (4)

As the action is explicitly metric independent, the corresponding partition
function is a topological invariant of the three-manifold $M$. The other
topological invariants associated with links can be described by expectation
value of Wilson loop operators in Chern-Simons theory. For an oriented knot
$\mathcal{K}$ in an irreducible representation $\mathcal{R}$, the Wilson loop
operator is defined as

$\displaystyle W_{\mathcal{R}}(\mathcal{K})$
$\displaystyle=\bigg{(}\text{Tr}_{R}\mathbb{H}^{(A)}[\mathcal{K}]\bigg{)}\bigg{(}\text{Tr}_{n}\mathbb{H}^{(B)}[\mathcal{K}]\bigg{)}$
(5)

where,

$\displaystyle\mathbb{H}^{(A)}[\mathcal{K}]$
$\displaystyle=\mathcal{P}\bigg{[}\exp\oint_{\mathcal{K}}A\bigg{]}~{}~{}\text{and}~{}~{}\mathbb{H}^{(B)}[\mathcal{K}]=\mathcal{P}\bigg{[}\exp\oint_{\mathcal{K}}B\bigg{]}$
(6)

are the holonomies of $SU(N)$ and $U(1)$ gauge connection respectively. The
symbol $\mathcal{P}$ denotes path ordering. Note that the representation
$\mathcal{R}\in U(N)$ involves representation $R\in SU(N)$ and charge $n\in
U(1)$. Given $(R,n)$, we can define $U(N)$ representation $\mathcal{R}$ in the
following way:

$\displaystyle n_{i}$
$\displaystyle=\begin{cases}l_{i}+s,~{}~{}~{}~{}i=1,...,N-1\\\
s~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}i=N\end{cases}$ (7)

where $l_{i}$ ($n_{i}$) is the number of boxes in the $i^{th}$ row
corresponding to the Young diagram of $SU(N)$ ($U(N)$) representation $R$
($\mathcal{R}$) and $s$ is any integer. The total number of boxes of
representation $\mathcal{R}$, known as the charge of $U(1)$ theory, is given
by $n=\sum_{i=1}^{N}n_{i}$. The relation between $SU(N)$ Chern-Simons theory
to $\mathfrak{su}(N)_{k}$ Wess-Zumino Witten conformal field theory implies
that the $l_{i}$’s of the $SU(N)$ representations must obey $l_{i}\leq k$.
Such representations are known as integrable representations. With suitable
choice of $s$, we can shift the range of $n_{i}$ appropriately.

If instead we have a link $\mathcal{L}$ made up of $L$ number of component
knots $\mathcal{K}_{a}$ each carrying representation $\mathcal{R}_{a}$ of
$U(N)$ gauge group, then the Wilson loop operator of $\mathcal{L}$ is

$\displaystyle W_{\mathcal{R}_{1},\cdots\mathcal{R}_{L}}[\mathcal{L}]$
$\displaystyle=\prod_{a=1}^{L}W_{\mathcal{R}_{a}}[\mathcal{K}_{a}].$ (8)

The correlation function of this topological operator yields the link
invariant of $\mathcal{L}$ corresponding to $U(N)$ Chern-Simons theory

$\displaystyle\mathcal{W}^{\mathcal{L}}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}({M},k,k_{1})$
$\displaystyle=\langle{W}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}[\mathcal{L}]\rangle=\frac{1}{Z(M)}\int_{M}[\mathcal{D}A][\mathcal{D}B]\bigg{(}\prod_{a=1}^{L}{W}_{\mathcal{R}_{a}}[\mathcal{K}_{a}]\bigg{)}e^{iS}$
$\displaystyle=\mathcal{W}^{\mathcal{L}}_{n^{(1)},\cdots,n^{(L)}}(M,k_{1})\mathcal{W}^{\mathcal{L}}_{R_{1},\cdots,R_{L}}(M,k).$
(9)

These link invariants can be computed using one of the two framings:
$\bullet$ standard framing where the self-linking numbers of the component
knots are zero,
$\bullet$ vertical framing where the self-linking number of any component knot
is equal to its crossing number.

In fact, the link invariants in standard framing are unchanged under all the
three Reidemeister moves and are called ambient isotopy invariants. However
the link invariants in vertical framing, referred to as framed link
invariants, are unchanged only under the Reidemeister moves II and III. These
framed link invariants are known in the knot theory literature as regular
isotopy invariants.

In this work, we will confine ourselves to framed links [$\mathcal{L};f$]
where the framing $f=\\{f_{1},\cdots,f_{L}\\}$ is a set of $L$ integers, each
element of the set denoting the self-linking number (or framing number) of the
component knot.

Let us consider the three manifold $M$ to be $S^{3}$. The $U(1)$ invariant of
$[\mathcal{L};f]$ is given as

$\displaystyle\mathcal{W}^{[\mathcal{L};f]}_{n^{(1)},\cdots,n^{(L)}}(S^{3},k_{1})$
$\displaystyle=\exp\bigg(\frac{i\pi}{k_{1}}\sum_{a=1}^{L}f_{a}\big{(}n^{(a)}\big{)}^{2}\bigg{missing})\exp\bigg(\frac{i\pi}{k_{1}}\sum_{a\neq
b}^{L}(lk_{ab})n^{(a)}n^{(b)}\bigg{missing})$ (10)

where $lk_{ab}$ denotes the linking number between the components
$\mathcal{K}_{a}$ and $\mathcal{K}_{b}$. For the polynomial invariant of
$U(N)$ Chern-Simons theory to be a function only of two variables

$\displaystyle q=\exp\left(\frac{2\pi i}{k+N}\right)~{}\text{and}~{}v=q^{N},$
(11)

in the $U(1)$ invariant (10) we need to make the following replacement for the
charge $n^{(a)}$ and the coupling constant $k_{1}$ borhade2004u ;
marino2001framed

$\displaystyle
n^{(a)}\rightarrow\frac{n^{(a)}}{\sqrt{N}}~{}~{},~{}~{}k_{1}\rightarrow k+N.$
(12)

We will elaborate the salient features of computing the $(L\alpha,L\beta)$
torus link invariants in the following section.

##  3 Torus link invariants in $U(N)$ Chern-Simons theory

Torus links with $L$ components are characterised by two integers
$(L\alpha,L\beta)$ where $\alpha$ and $\beta$ are coprime to each other. These
links can be obtained from the closure of a braid with $L\alpha$ strands. The
braid word for a right handed $(L\alpha,L\beta)$ torus link, in terms of the
braid generators $b_{i}$, is

$\displaystyle\mathcal{B}(L\alpha,L\beta)$
$\displaystyle=\left(b_{1}b_{2}...b_{L\alpha-1}\right)^{L\beta}.$ (13)

The corresponding mirror image of this link is left handed and its braid word
becomes $\left(b^{-1}_{1}b^{-1}_{2}...b^{-1}_{L\alpha-1}\right)^{L\beta}$. As
an illustration, we show in figure (1) a two component ($L=2$) torus link
(with $\alpha=1,~{}\beta=m$) and its mirror image , where $m$ is any positive
integer. The dotted lines in this figure indicate the closure of the two-
strand braid.

(a) Right handed
(linking number +$m$)

(b) Left handed
(linking number -$m$)

Figure 1: $(2,2m)$ torus link: closure of two parallely oriented strands with
$2m$ crossings

For clarity, we now discuss the colored HOMFLY-PT polynomial for these
$(2,2m)$ torus links before getting into arbitrary $L$ component torus links.

###  3.1 $(2,2m)$ torus link invariant

Given a braid word (13), there is a systematic procedure of directly writing
down the invariants of links that are obtained from the closure of such braids
RamaDevi:1992np . Basically, we use the basis where the braiding generators
are diagonal. As any two consecutive generators $b_{i}$ and $b_{i\pm 1}$ do
not commute, we need to perform a suitable unitary transformation to go from
one basis to another. The invariant of the right-handed $(2,2m)$ torus link
(refer to Fig. 1(a)), closure of a simple two-strand braid with braid word
$b_{1}^{2m}$, involves the eigenvalues only of the braiding generator $b_{1}$.
In vertical framing, the eigenvalues for the right handed braiding generator
$b_{1}$ on two parallely oriented strands carrying representations ${R}_{1}$
and ${R}_{2}$ of $SU(N)$ gauge group is given by kaul2001three

$\displaystyle\lambda_{R_{t}}^{(+)}(R_{1},R_{2})$
$\displaystyle=(-1)^{\epsilon_{R_{1}}+\epsilon_{R_{2}}-\epsilon_{R_{t}}}q^{-\big{(}\frac{C_{R_{1}}+C_{R_{2}}}{2}\big{)}+\frac{C_{R_{t}}}{2}},~{}~{}~{}~{}~{}R_{t}\in
R_{1}\otimes R_{2}.$ (14)

where $\epsilon_{R_{1}},\epsilon_{R_{2}},\epsilon_{R_{t}}=\pm 1$, $q$ is
defined in equation (11) and $C_{R_{a}}$, the quadratic Casimir of
representation $R_{a}$ is defined as

$\displaystyle C_{R_{a}}$
$\displaystyle=-\frac{\big{(}l^{(a)}\big{)}^{2}}{2N}+\frac{1}{2}\sum_{j}l_{j}^{(a)}\big{(}l_{j}^{(a)}-2j+N+1\big{)}.$
(15)

Here $l_{i}^{(a)}$ denotes the number of boxes in the $i$th row of
representation $R_{a}$ and $l^{(a)}=\sum_{i}l_{i}^{(a)}$. Using the above
braiding eigenvalue (14), we can write down the colored HOMFLY-PT polynomial
of $(2,2m)$ torus link with framing $f=\\{f_{1},f_{2}\\}$ of the component
knots and linking number $m$ borhade2004u :

$\displaystyle\mathcal{W}_{R_{1},R_{2}}^{[(2,2m);f]}(S^{3},k)$
$\displaystyle=q^{f_{1}C_{R_{1}}+f_{2}C_{R_{2}}}\sum_{R_{t}\in R_{1}\otimes
R_{2}}\big{(}\text{dim}_{q}R_{t}\big{)}\bigg{(}\lambda^{(+)}_{{R}_{t}}\bigg{)}^{2m}$
$\displaystyle=q^{f_{1}C_{R_{1}}+f_{2}C_{R_{2}}}\sum_{R_{t}\in R_{1}\otimes
R_{2}}(\text{dim}_{q}R_{t})\bigg{(}q^{-\frac{C_{R_{1}}+C_{R_{2}}}{2}+\frac{C_{R_{t}}}{2}}\bigg{)}^{2m},$
(16)

where the $q$-dimension of a representation is defined as

$\displaystyle\text{dim}_{q}R_{a}$ $\displaystyle=\prod_{1\leq i<j\leq
N}\frac{\big{[}l_{i}^{(a)}-l_{j}^{(a)}+j-i\big{]}_{q}}{[j-i]_{q}}~{}~{}\text{and}~{}~{}[x]_{q}=\frac{q^{\frac{x}{2}}-q^{-\frac{x}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}.$
(17)

The $U(N)$ invariant of framed $(2,2m)$ torus link can be written using (9,
10, 16):

$\displaystyle\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}^{[(2,2m);f]}(S^{3},k)$
$\displaystyle=\mathcal{W}_{n^{(1)},n^{(2)}}^{[(2,2m);f]}(S^{3},k)\times\mathcal{W}_{R_{1},R_{2}}^{[(2,2m);f]}(S^{3},k)$
$\displaystyle=\exp\bigg(\frac{i\pi}{k+N}\sum_{a=1}^{2}\frac{f_{a}\big{(}n^{(a)}\big{)}^{2}}{N}\bigg{missing})\exp\bigg(\frac{i\pi}{k+N}\frac{2(+m)n^{(1)}n^{(2)}}{N}\bigg{missing})$
$\displaystyle~{}~{}~{}\times
q^{f_{1}C_{\mathcal{R}_{1}}+f_{2}C_{\mathcal{R}_{2}}}\sum_{\mathcal{R}_{t}\in\mathcal{R}_{1}\otimes\mathcal{R}_{2}}(\text{dim}_{q}\mathcal{R}_{t})\bigg{(}q^{-\frac{C_{\mathcal{R}_{1}}+C_{\mathcal{R}_{2}}}{2}+\frac{C_{\mathcal{R}_{t}}}{2}}\bigg{)}^{2m}.$
(18)

Note that, in writing the above expression we have used the fact that the
quadratic Casimir and $q$-dimension of a $U(N)$ representation
$\mathcal{R}_{a}$ is same as that of the quadratic Casimir and $q$-dimension
of $SU(N)$ representation $R_{a}$:

$\displaystyle C_{\mathcal{R}_{a}}$
$\displaystyle=-\frac{\big{(}n^{(a)}\big{)}^{2}}{2N}+\frac{1}{2}\sum_{i=1}^{N}n_{i}^{(a)}\big{(}n_{i}^{(a)}-2i+N+1\big{)}=C_{R_{a}}.$
(19) $\displaystyle\text{dim}_{q}$ $\displaystyle\mathcal{R}_{a}=\prod_{1\leq
i<j\leq
N}\frac{\big{[}n_{i}^{(a)}-n_{j}^{(a)}+j-i\big{]}_{q}}{\big{[}j-i\big{]}_{q}}=\text{dim}_{q}R_{a}.$
(20)

We introduce $\kappa_{\mathcal{R}_{a}}$ in terms of the number of boxes
$n_{i}^{(a)}$ in the $i$th row of the Young diagram of representation
$\mathcal{R}_{a}$:

$\kappa_{\mathcal{R}_{a}}=\frac{1}{2}\sum_{i=1}^{N}n_{i}^{(a)}\big{(}n_{i}^{(a)}-2i+N+1\big{)}.$
(21)

In terms of these variables, we express the $U(N)$ invariant of the framed
$(2,2m)$ torus link as

$\displaystyle\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}^{[(2,2m);f]}(S^{3},k)$
$\displaystyle=q^{f_{1}\kappa_{\mathcal{R}_{1}}+f_{2}\kappa_{\mathcal{R}_{2}}}\sum_{\mathcal{R}_{t}}(\text{dim}_{q}\mathcal{R}_{t})\bigg{(}q^{-\kappa_{\mathcal{R}_{1}}-\kappa_{\mathcal{R}_{2}}+\kappa_{\mathcal{R}_{t}}}\bigg{)}^{m}.$
(22)

Hereafter we refer $\kappa_{\mathcal{R}_{a}}$ to be the quadratic Casimir of
$U(N)$ representation $\mathcal{R}_{a}$. The invariants for arbitrary
$L$-component torus links can be compactly written using these quadratic
casimirs $\kappa_{\mathcal{R}_{a}}$ and $q$-dimension of representations
lin2010hecke . We will briefly present the explicit form of these link
invariants in the following subsection.

###  3.2 $(L\alpha,L\beta)$ torus link invariant

For a general two-component right handed torus link $(2\alpha,2\beta)$
carrying $U(N)$ representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ on the
component knots, the polynomial invariant is lin2010hecke

$\displaystyle\mathcal{W}^{(2\alpha,2\beta)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3},k)$
$\displaystyle=\prod_{i=1}^{2}\left(q^{-\frac{\alpha\beta}{2}\varkappa_{\mathcal{R}_{i}}}~{}v^{-\frac{\beta(\alpha-1)}{2}n^{(i)}}\right)\sum_{\mathcal{R}_{t}\vdash\alpha(n^{(1)}+n^{(2)})}\zeta^{\mathcal{R}_{t}}_{\mathcal{R}_{1},\mathcal{R}_{2}}q^{\frac{\beta}{2\alpha}\varkappa_{\mathcal{R}_{t}}}s^{\star}_{\mathcal{R}_{t}}(q,v)$
(23)

where

$\displaystyle\zeta^{\mathcal{R}_{t}}_{\mathcal{R}_{1},\mathcal{R}_{2}}$
$\displaystyle=\sum_{\Lambda^{(1)}\vdash
n^{(1)}}\frac{|c_{\Lambda^{(1)}}|}{(n^{(1)})!}\chi_{\mathcal{R}_{1}}(c_{\Lambda^{(1)}})\sum_{\Lambda^{(2)}\vdash
n^{(2)}}\frac{|c_{\Lambda^{(2)}}|}{(n^{(2)})!}\chi_{\mathcal{R}_{2}}(c_{\Lambda^{(2)}})\chi_{\mathcal{R}_{t}}(c_{(\Lambda^{(1)}+\Lambda^{(2)})_{(\alpha)}}),$
$\displaystyle s_{\mathcal{R}_{t}}^{\star}(q,$ $\displaystyle
v)=\sum_{\Lambda\vdash
n^{(t)}}\frac{|c_{\Lambda}|}{(n^{(t)})!}\chi_{\mathcal{R}_{t}}(c_{\Lambda})\prod_{i=1}^{\ell(\Lambda)}\frac{v^{\Lambda_{i}/2}-v^{-\Lambda_{i}/2}}{q^{\Lambda_{i}/2}-q^{-\Lambda_{i}/2}}\equiv\text{dim}_{q}\mathcal{R}_{t},~{}~{}v=q^{N}.$
(24)

Here $\varkappa_{\mathcal{R}}$ in terms of $\kappa_{\mathcal{R}}$ (21) can be
written as

$\displaystyle\varkappa_{\mathcal{R}}=\sum_{i}n_{i}(n_{i}-2i+1)=2\kappa_{\mathcal{R}}-N\sum_{i}n_{i}.$
(25)

The notation $\Lambda^{(a)}\vdash n^{(a)}$ means that
$\Lambda^{(a)}\equiv\left(\Lambda^{(a)}_{1},\Lambda^{(a)}_{2},\Lambda^{(a)}_{3},...\right)$
is a partition, in other words a sequence of positive integers with
$\Lambda_{1}^{(a)}\geq\Lambda_{2}^{(a)}\geq\Lambda_{3}^{(a)}\geq\cdots$,
satisfying

$\displaystyle\sum_{i=1}^{\ell(\Lambda^{(a)})}\Lambda^{(a)}_{i}=n^{(a)},$ (26)

where $\ell(\Lambda^{(a)})$ is the highest value of $i$ for which
$\Lambda^{(a)}_{i}$ is non zero:

$\displaystyle\ell(\Lambda^{(a)})$
$\displaystyle=\mathrm{max}\left\\{i|\Lambda^{(a)}_{i}>0\right\\}.$ (27)

The conjugacy class associated to the partition $\Lambda^{(a)}$ is denoted as
$c_{\Lambda^{(a)}}$ and it consists of one $\Lambda^{(a)}_{1}$-cycle, one
$\Lambda^{(a)}_{2}$-cycle, one $\Lambda^{(a)}_{3}$-cycle and so on.
$|c_{\Lambda^{(a)}}|$ denotes the total number of elements belonging to the
conjugacy class $c_{\Lambda^{(a)}}$, having the same cycle structure.
$\chi_{\mathcal{R}_{a}}(c_{\Lambda^{(a)}})$ is the corresponding character
labelled by representation $\mathcal{R}_{a}$. The sum of two partitions is the
sum of each of the elements of the respective partitions

$\displaystyle\Lambda^{(1)}+\Lambda^{(2)}\equiv\left(\Lambda^{(1)}_{1}+\Lambda^{(2)}_{1},\Lambda^{(1)}_{2}+\Lambda^{(2)}_{2},\Lambda^{(1)}_{3}+\Lambda^{(2)}_{3},\cdots\right),$
(28)

and $(\Lambda)_{(\alpha)}$ implies the partition
$(\alpha\Lambda_{1},\alpha\Lambda_{2},\alpha\Lambda_{3},...)$.

Rewriting the torus link invariant (23) in terms of variable
$\kappa_{\mathcal{R}}$ we get:

$\displaystyle\mathcal{W}^{(2\alpha,2\beta)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3},k)$
$\displaystyle=q^{-{\alpha\beta}\big{(}\kappa_{\mathcal{R}_{1}}+\kappa_{\mathcal{R}_{2}}\big{)}}\sum_{\mathcal{R}_{t}\vdash\alpha(n^{(1)}+n^{(2)})}\zeta^{\mathcal{R}_{t}}_{\mathcal{R}_{1},\mathcal{R}_{2}}q^{\frac{\beta}{\alpha}\kappa_{\mathcal{R}_{t}}}s^{\star}_{\mathcal{R}_{t}}(q,v).$
(29)

If the component knots are having $f_{1}$ and $f_{2}$ self-linking numbers,
which we denote as $f=\\{f_{1},f_{2}\\}$, then the polynomial invariant will
have an additional framing factor

$\displaystyle\mathcal{W}^{[(2\alpha,2\beta);f]}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3},k)$
$\displaystyle=q^{f_{1}\kappa_{\mathcal{R}_{1}}+f_{2}\kappa_{\mathcal{R}_{2}}}\mathcal{W}^{(2\alpha,2\beta)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3},k).$
(30)

Now, we can generalise the colored HOMFLY-PT polynomial for an arbitrary $L$
component torus link $(L\alpha,L\beta)$, with self-linking number
$f=\\{f_{1},...f_{L}\\}$ of the component knots, as

$\displaystyle\mathcal{W}^{[(L\alpha,L\beta);f]}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}(S^{3},k)=\bigg{(}\prod_{a=1}^{L}$
$\displaystyle
q^{(f_{a}-\alpha\beta)\kappa_{\mathcal{R}_{a}}}\bigg{)}\sum_{\mathcal{R}_{t}\vdash\alpha(n^{(1)}+\cdots+n^{(L)})}\zeta^{\mathcal{R}_{t}}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}q^{\frac{\beta}{\alpha}\kappa_{\mathcal{R}_{t}}}s^{\star}_{\mathcal{R}_{t}}(q,v)$
(31)

where

$\displaystyle\zeta^{\mathcal{R}_{t}}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}=\prod_{a=1}^{L}\left(\sum_{\Lambda^{(a)}\vdash
n^{(a)}}\frac{|c_{\Lambda^{(a)}}|}{(n^{(a)})!}\chi_{\mathcal{R}_{a}}(c_{\Lambda^{(a)}})\right)\chi_{\mathcal{R}_{t}}(c_{(\Lambda^{(1)}+\cdots+\Lambda^{(L)})_{(\alpha)}}).$
(32)

In Ref. chakraborty2022large , two of the authors have analysed the double
scaling limit (1,2) of the two-component $(2,2m)$ torus link invariants.
Particularly, these invariants were shown to be expressed in terms of $(2,2)$
torus link invariants. We believe that this must be generalisable for an
arbitrary $L$-component torus link $(L\alpha,L\beta)$. In the following
section we present our result after briefly reviewing the works on $(2,2m)$
torus links.

##  4 Torus link invariants in the double scaling limit of $U(N)$ Chern-Simons
theory

As discussed in the introduction, the double scaling limit (1) of $U(N)$
Chern-Simons theory involves the Chern-Simons coupling $k\rightarrow\infty$.
The range of the $n^{(a)}_{i}$’s (7) for the integrable representations
$\mathcal{R}_{a}$ at large $k$ can be chosen as

$-\frac{k}{2}\leq n_{N}^{(a)}\leq\cdots\leq n_{1}^{(a)}\leq\frac{k}{2}.$ (33)

For the representation $\mathcal{R}_{a}$ we define a set of $N$ variables
$\\{\theta_{1}^{(a)},...,\theta_{N}^{(a)}\\}$:

$\displaystyle\theta_{i}^{(a)}$
$\displaystyle=\frac{2\pi}{N+k}\bigg{(}h_{i}^{(a)}-\frac{N-1}{2}\bigg{)},$
(34) $\displaystyle\text{where,}\quad h_{i}^{(a)}$
$\displaystyle=n_{i}^{(a)}+N-i.$ (35)

It is easy to check that in the double scaling limit the variables
$\theta_{i}^{(a)}$ range from $-\pi$ to $\pi$. The quadratic Casimir of
representation $\mathcal{R}_{a}$ (21) can be expressed in terms of these
variables as

$\displaystyle\kappa_{\mathcal{R}_{a}}$
$\displaystyle=\frac{(N+k)^{2}}{8\pi^{2}}\sum_{i}{\theta_{i}^{(a)}}^{2}-\frac{N(N^{2}-1)}{24}.$
(36)

In the large $N$ limit, the set of discrete variables $\theta_{i}$ becomes a
continuous function

$\displaystyle\theta^{(a)}_{i}\rightarrow\theta^{(a)}(x)$
$\displaystyle~{}~{}~{}~{}\text{where}~{}~{}x=\frac{i}{N}\in[0,1].$ (37)

All the discrete summations over $i$ are then replaced by integration over $x$

$\frac{1}{N}\sum_{i}=\int_{0}^{1}dx.$ (38)

Moreover the distribution of $\theta_{i}$, in the large $N$ limit is captured
by a distribution function

$\sigma(\theta)=\frac{1}{N}\sum_{i=1}^{N}\delta(\theta-\theta_{i})=\left|\frac{\partial
x}{\partial\theta}\right|,$ (39)

such that $\int d\theta\sigma(\theta)=1$. Hence, for any arbitrary functions
of $\theta_{i}$ we can write

$\displaystyle\frac{1}{N}\sum_{i}f(\theta_{i})\rightarrow\int_{0}^{1}dxf(\theta(x))=\int
d\theta\ \sigma(\theta)f(\theta).$ (40)

Now we will incorporate the continuum limit discussed above for the $(2,2m)$
torus link invariant.

###  4.1 ($2,2m$) torus link invariant in the double scaling limit

The invariant
$\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}^{[(2,2m);f]}(S^{3},k)$ (22) can
be expressed using the $\\{\theta_{i}\\}$ variables (34) as

$\displaystyle\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}^{[(2,2m);f]}(S^{3}$
$\displaystyle,k)=\exp[\frac{i(N+k)}{4\pi}\bigg{(}(f_{1}-m)\sum_{i}{\theta_{i}^{(1)}}^{2}+(f_{2}-m)\sum_{i}{\theta_{i}^{(2)}}^{2}\bigg{)}\bigg{]}$
(41) $\displaystyle\times\exp[-$
$\displaystyle\frac{i\pi(N^{3}-N)}{12(N+k)}(f_{1}+f_{2}-m)\bigg{]}\times\sum_{\\{\theta^{(t)}_{i}\\}}\frac{\textstyle\exp[\frac{1}{2}\sum_{i\neq
j}^{N}\log\left|\sin\left(\frac{\theta^{(t)}_{i}-\theta^{(t)}_{j}}{2}\right)\right|\bigg{]}}{\textstyle\exp[\frac{1}{2}\sum_{i\neq
j}^{N}\log~{}\left|\sin\left(\frac{\pi(j-i)}{N+k}\right)\right|\bigg{]}}$
$\displaystyle\times\exp[\frac{im(N+k)}{4\pi}\sum_{i}{\theta^{(t)}_{i}}^{2}\bigg{]}.$

Here $\theta_{i}^{(1)}$, $\theta_{i}^{(2)}$ and $\theta_{i}^{(t)}$ are the set
of variables corresponding to representations $\mathcal{R}_{1}$,
$\mathcal{R}_{2}$ and $\mathcal{R}_{t}$ respectively. In the double scaling
limit with appropriate replacement of the discrete variables in terms of the
continuous functions (37, 38, 39) the above invariant becomes

$\displaystyle{\mathcal{W}}^{[(2,2m);f]}$
$\displaystyle(S^{3},\lambda)={\mathcal{G}}^{(f)}(m,\lambda,\sigma_{1},\sigma_{2})\mathcal{F}(m,\lambda)$
(42)

where

$\displaystyle{\mathcal{G}}^{(f)}(m,$
$\displaystyle\lambda,\sigma_{1},\sigma_{2})=\exp[\frac{iN^{2}}{4\pi\lambda}\bigg{(}\int
d\theta\theta^{2}\left(\left(f_{1}-m\right){\sigma_{1}(\theta)}+\left(f_{2}-m\right){\sigma_{2}(\theta)}\right)\bigg{)}\bigg{]}$
(43) $\displaystyle~{}~{}~{}~{}~{}~{}\times\exp[-\frac{i\pi\lambda
N^{2}}{12}(f_{1}+f_{2}-m)-\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy\log\left|\sin\left(\pi\lambda(y-x)\right)\right|\bigg{]}$

is dependent on framing $f$ and

$\displaystyle\mathcal{F}(m,\lambda)$
$\displaystyle=\int[\mathcal{D}\theta^{(t)}(x)]\exp[\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy~{}\log~{}\bigg{|}\sin\left(\frac{\theta^{(t)}(x)-\theta^{(t)}(y)}{2}\right)\bigg{|}\bigg{]}$
(44)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp[\frac{imN^{2}}{4\pi\lambda}\int
dx~{}\theta^{(t)}(x)^{2}\bigg{]}$

is independent of framing. Observe that the function
${\mathcal{G}^{(f)}}(m,\lambda,\sigma_{1},\sigma_{2})$ depends only on the
given representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ and is
independent of $\mathcal{R}_{t}$. However, $\mathcal{F}(m,\lambda)$ involves
the $\theta^{(t)}$ integration which is constrained, since it runs only over
the irreducible representations $\mathcal{R}_{t}$ appearing in the tensor
product of $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$. Therefore, this function
$\mathcal{F}(m,\lambda)$ is difficult to evaluate even in the large $N$ limit.
Note that, the space of functional integration over $\theta^{(t)}(x)$ in
$\mathcal{F}(m,\lambda)$ remains the same for any value of $m$. As a result,
the function $\mathcal{F}(m,\lambda)$ has the following property

$\mathcal{F}(m,\lambda)=\mathcal{F}(1,\lambda/m).$ (45)

With $m=1$,
$\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}^{[(2,2m);f]}(S^{3},k)$ becomes
the invariant of framed Hopf link. Therefore, by exploiting the property (45),
we can see that a framed $(2,2m)$ torus link invariant can be expressed in
terms of a framed Hopf link invariant as follows:

$\displaystyle{\mathcal{W}}_{\sigma_{1},\sigma_{2}}^{[(2,2m);f]}(S^{3},\lambda)$
$\displaystyle=\bigg{(}\frac{\mathcal{G}^{(f)}(m,\lambda,\sigma_{1},\sigma_{2})}{\mathcal{G}^{(f)}(1,\lambda/m,\sigma_{1},\sigma_{2})}\bigg{)}~{}\mathcal{W}_{\sigma_{1},\sigma_{2}}^{[(2,2);f]}(S^{3},\lambda/m).$
(46)

That is, we first find the distribution functions $\sigma_{1},\sigma_{2}$
corresponding to the representations $\mathcal{R}_{1}$, $\mathcal{R}_{2}$
placed on the two component knots. After which we can write the Hopf link
invariant as a function of $\sigma_{1}$, $\sigma_{2}$ and $\lambda$. Then the
link invariant for $(2,2m)$ torus link as a function of $\lambda$ becomes
proportional to the Hopf link invariant with $\lambda$ replaced by $\lambda/m$
but importantly, keeping $\sigma_{1}$ and $\sigma_{2}$ fixed while performing
such a replacement.

In the following subsection we generalise (46) for any $L$ component torus
link $(L\alpha,L\beta)$ embedded inside $S^{3}$.

###  4.2 $(L\alpha,L\beta)$ torus link invariant in the double scaling limit

Let us first focus on arbitrary 2 component torus links $(2\alpha,2\beta)$.
Using the expression of $\kappa_{\mathcal{R}_{a}}$ in terms of the set of $N$
variables $\theta_{i}^{(a)}$ (36) we can express the $(2\alpha,2\beta)$ torus
link invariant (30) as

$\displaystyle\mathcal{W}^{[(2\alpha,2\beta);f]}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3},k)=\exp[\frac{i(N+{k})}{4\pi}\bigg{(}(f_{1}-\alpha\beta)\sum_{i}{\theta_{i}^{{(1)}}}^{2}+(f_{2}-\alpha\beta)\sum_{i}{\theta_{i}^{{(2)}}}^{2}\bigg{)}\bigg{]}$
(47)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp[-\frac{i\pi(N^{3}-N)}{12(N+{k})}(f_{1}+f_{2}-2\alpha\beta+\frac{\beta}{\alpha})\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}\times\sum_{\\{\theta^{(t)}_{i}\\}}\zeta^{\theta^{(t)}}_{\theta^{{(1)}},\theta^{{(2)}}}\exp[\frac{i\beta(N+{k})}{4\pi\alpha}\sum_{i}{\theta^{(t)}_{i}}^{2}\bigg{]}\frac{\textstyle\exp[\frac{1}{2}\sum_{i\neq
j}^{N}\log\left|\sin\left(\frac{\theta^{(t)}_{i}-\theta^{(t)}_{j}}{2}\right)\right|\bigg{]}}{\textstyle\exp[\frac{1}{2}\sum_{i\neq
j}^{N}\log~{}\left|\sin\left(\frac{\pi(j-i)}{N+k}\right)\right|\bigg{]}}.$

We now take the double scaling limit with the appropriate replacement of the
discrete variables in terms of the continuous functions (37, 38, 39):

$\displaystyle\mathcal{W}^{[(2\alpha,2\beta);f]}_{{\sigma}_{1},{\sigma}_{2}}(S^{3},\lambda)=\exp[\frac{iN^{2}}{4\pi\lambda}\bigg{(}\int
d\theta\theta^{2}\big{(}(f_{1}-\alpha\beta)\sigma_{1}(\theta)+(f_{2}-\alpha\beta)\sigma_{2}(\theta)\big{)}\bigg{)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp[-\frac{i\pi\lambda
N^{2}}{12}(f_{1}+f_{2}-2\alpha\beta+\frac{\beta}{\alpha})\bigg{]}$ (48)
$\displaystyle\times$
$\displaystyle\int[\mathcal{D}\theta^{(t)}]\zeta^{\theta^{(t)}}_{\theta^{(1)},\theta^{(2)}}\exp[\frac{i\beta
N^{2}}{4\pi\alpha\lambda}\int
d\theta\theta^{2}\sigma_{t}(\theta)\bigg{]}\frac{\textstyle\exp[\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy\log|\sin\left(\frac{\theta^{(t)}(x)-\theta^{(t)}(y)}{2}\right)\bigg{|}\bigg{]}}{\textstyle\exp[\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy\log|\sin\left({\pi\lambda(y-x)}\right)\big{|}\bigg{]}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\equiv{\mathcal{G}}^{(f)}(\alpha,\beta,\lambda,\sigma_{1},\sigma_{2}){\mathcal{F}}\bigg{(}\frac{\beta}{\alpha},\lambda\bigg{)},$
(49)

where

$\displaystyle{\mathcal{G}}^{(f)}(\alpha,\beta,\lambda,\sigma_{1},\sigma_{2})=\exp[\frac{iN^{2}}{4\pi\lambda}\bigg{(}\int
d\theta\theta^{2}\big{(}(f_{1}-\alpha\beta)\sigma_{1}(\theta)+(f_{2}-\alpha\beta)\sigma_{2}(\theta)\big{)}\bigg{)}\bigg{]}$
(50) $\displaystyle~{}~{}\times\exp[-\frac{i\pi\lambda
N^{2}}{12}\left(f_{1}+f_{2}-2\alpha\beta+\frac{\beta}{\alpha}\right)-\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy~{}\log|\sin\left(\pi\lambda(y-x)\right)\big{|}\bigg{]},$

and

$\displaystyle{\mathcal{F}}\bigg{(}\frac{\beta}{\alpha},\lambda\bigg{)}=\int[\mathcal{D}\theta^{(t)}]\zeta^{\theta^{(t)}}_{\theta^{(1)},\theta^{(2)}}\exp[\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int~{}dy~{}\log~{}\bigg{|}\sin\left(\frac{\theta^{(t)}(x)-\theta^{(t)}(y)}{2}\right)\bigg{|}\bigg{]}$
(51)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp[\frac{i\beta
N^{2}}{4\pi\alpha\lambda}\int d\theta\theta^{2}\sigma_{t}(\theta)\bigg{]}.$

When $\alpha=\beta=1$, we get the double scaling limit of Hopf link invariant.
We exploit the symmetry of
${\mathcal{F}}\bigg{(}\frac{\beta}{\alpha},\lambda\bigg{)}$ under

$\displaystyle\frac{\beta}{\alpha}\rightarrow
1~{}~{}\text{and}~{}~{}\lambda\rightarrow\frac{\lambda}{\beta/\alpha}$ (52)

to write down the invariant of any 2 component torus link in terms of Hopf
link invariant as

$\displaystyle\mathcal{W}^{[(2\alpha,2\beta);f]}_{{\sigma}_{1},{\sigma}_{2}}$
$\displaystyle(S^{3},\lambda)=\frac{{\widetilde{\mathcal{G}}}^{(f)}(\alpha,\beta,\lambda,\sigma_{1},\sigma_{2})}{\widetilde{\mathcal{G}}^{(f)}(1,1,\frac{\lambda}{\beta/\alpha},\sigma_{1},\sigma_{2})}\mathcal{W}^{[(2,2);f]}_{{\sigma}_{1},{\sigma}_{2}}\bigg{(}S^{3},\frac{\lambda}{\beta/\alpha}\bigg{)}.$
(53)

Similar exercise for any arbitrary $L$ component torus link (31) is
straightforward and we find that $(L\alpha,L\beta)$ torus link invariant can
be expressed in terms of $(L,L)$ torus link invariant as follows:

$\displaystyle\mathcal{W}^{[(L\alpha,L\beta);\\{f_{1},...,f_{L}\\}]}_{{\sigma}_{1},...,{\sigma}_{L}}$
$\displaystyle(S^{3},\lambda)=\frac{\widetilde{\mathcal{G}}^{(f)}(\alpha,\beta,\lambda,\sigma_{1},...,\sigma_{L})}{\widetilde{\mathcal{G}}^{(f)}(1,1,\frac{\lambda}{\beta/\alpha},\sigma_{1},...,\sigma_{L})}\mathcal{W}^{[(L,L);\\{f_{1},...,f_{L}\\}]}_{{\sigma}_{1},...{\sigma}_{L}}\bigg{(}S^{3},\frac{\lambda}{\beta/\alpha}\bigg{)}.$
(54)

Thus, given a $(L,L)$ torus link invariant in variable $\lambda$ with
$\sigma_{1},...,\sigma_{L}$ being the eigenvalue densities of the component
knots, the $(L\alpha,L\beta)$ torus link invariant (carrying the same
eigenvalue densities) in variable $\lambda$ becomes proportional to the
$(L,L)$ torus link invariant with $\lambda$ replaced by
$\frac{\lambda}{\beta/\alpha}$ while keeping $\sigma_{1},...,\sigma_{L}$
unaltered.

We can explicitly work out the $(L,L)$ link invariants (31) with components
carrying different representations whose Young diagrams have small number of
boxes. For these representations, we can infer the leading and subleading
terms for a general $(L\alpha,L\beta)$ link invariant in the double scaling
limit (54). Extracting such leading and subleading contributions appear to be
practically difficult for representations with large number of boxes. In Ref.
chakraborty2022large , large $N$ contributions for the two component Hopf link
invariant was determined by mapping it to a one dimensional fluid equation,
where the initial and final fluid densities correspond to two of the
representations placed on the two component knots. However, such a procedure
is not generalisable for a $(L,L)$ torus link with $L>2$. Matrix model method
appears to be an efficient approach to handle such difficulties. In the
following section, we will briefly review $U(N)$ matrix model and the
computation of the correlators corresponding to the torus link invariants in
the double scaling limit.

##  5 Torus link invariants and matrix models at large $N$

Study of the partition function and the correlators in $U(N)$ matrix model at
large $N$ will allow us to deduce the leading and subleading contributions of
torus link invariants. Within the matrix model approach, we obtain analytic
results for these contributions at the following two instances:

  * •

representations having small number of boxes (small as compared to
$N,k\rightarrow\infty$), these are validated with those obtained from explicit
colored HOMLFLY-PT polynomials.

  * •

large symmetric representations placed on the component knots.

We now focus on the invariant of a framed Hopf link, embedded on a three-
manifold $S^{3}/\mathbb{Z}_{p}$, which can equivalently be written in terms of
the modular transformation matrices ($\mathcal{S},\mathcal{T}$) of
$\mathfrak{u}(N)_{k}$ Wess-Zumino conformal field theory. The explicit form of
Hopf link invariant on $S^{3}/\mathbb{Z}_{p}$ with representations
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ on the two components is given by

$\displaystyle\widetilde{{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}\left(S^{3}/\mathbb{Z}_{p},k\right)$
$\displaystyle=\sum_{\mathcal{R}}\mathcal{S}_{\mathcal{R}_{1}\mathcal{R}}\mathcal{S}_{\mathcal{R}_{2}\mathcal{R}}\mathcal{T}^{-p}_{\mathcal{R}\mathcal{R}}=\sum_{\mathcal{R}}\frac{\mathcal{S}_{\mathcal{R}_{1}\mathcal{R}}}{\mathcal{S}_{0\mathcal{R}}}\frac{\mathcal{S}_{\mathcal{R}_{2}\mathcal{R}}}{\mathcal{S}_{0\mathcal{R}}}\mathcal{S}_{0\mathcal{R}}^{2}\mathcal{T}^{-p}_{\mathcal{R}\mathcal{R}}.$
(55)

The summation runs over all the irreducible integrable representations
$\mathcal{R}$ of $\mathfrak{u}(N)_{k}$. Considering $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ to be trivial representations we obtain the partition
function of Chern-Simons theory on $S^{3}/\mathbb{Z}_{p}$

$\displaystyle\mathcal{Z}(S^{3}/\mathbb{Z}_{p},k)$
$\displaystyle=\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}.$
(56)

In the double scaling limit (1) the partition function is dominated by a
single representation. Contributions due to other representations are
suppressed by $\mathcal{O}(1/N^{2})$. This is the standard saddle point
approximation. The Hopf link invariant defined in (55) is unnormalised. We can
define the normalised invariant as

$\displaystyle{{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}\left(S^{3}/\mathbb{Z}_{p},k\right)=\frac{\widetilde{{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}\left(S^{3}/\mathbb{Z}_{p},k\right)}{\mathcal{Z}(S^{3}/\mathbb{Z}_{p},k)}$
(57)

such that
${{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}\left(S^{3}/\mathbb{Z}_{p},k\right)$
with the representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ being trivial
becomes identically equal to one. Note that, with $p=1$,
${\mathscr{V}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},k)$
becomes a Hopf link (22) embedded inside three manifold $S^{3}$ with framing
of component knots $f=\\{1,1\\}$.

In order to proceed further we explicitly write down the modular
transformation matrix $\mathcal{S}$ for ${\mathfrak{u}}(N)_{k}$

$\displaystyle\mathcal{S}_{\mathcal{R}_{a}\mathcal{R}_{b}}$
$\displaystyle=\frac{(-i)^{N(N-1)/2}}{(k+N)^{N/2}}e^{\frac{\textstyle 2\pi
in^{(a)}n^{(b)}}{\textstyle N(N+k)}}\det M(\mathcal{R}_{a},\mathcal{R}_{b}),$
(58)

where for any integrable representations $\mathcal{R}_{a}$ and
$\mathcal{R}_{b}$

$\displaystyle M_{ij}(\mathcal{R}_{a},\mathcal{R}_{b})$
$\displaystyle=\exp[\frac{2\pi
i}{k+N}\phi_{i}(\mathcal{R}_{a})\phi_{j}(\mathcal{R}_{b})\bigg{]},\quad
i,j=1,...,N$ (59) $\displaystyle\quad\phi_{i}(\mathcal{R}_{a})$
$\displaystyle=n_{i}^{(a)}-\frac{n^{(a)}}{N}-i+\frac{N+1}{2}.$ (60)

Here, as before, $n^{(a/b)}$ and $n^{(a/b)}_{i}$ denote the total number of
boxes and the number of boxes in the $i^{\text{th}}$ row corresponding to the
Young diagram of $\mathcal{R}_{a/b}$ respectively.

The other modular transformation matrix $\mathcal{T}$ is given by

$\mathcal{T}_{\mathcal{R}_{a}\mathcal{R}_{b}}=\exp[{2\pi
i\big{(}\mathcal{Q}_{\mathcal{R}_{a}}-\frac{c}{24}\big{)}}\bigg{]}\delta_{\mathcal{R}_{a}\mathcal{R}_{b}},$
(61)

$\mathcal{Q}_{\mathcal{R}}$ and $c$ denote the conformal weight of
representation $\mathcal{R}$ and central charge respectively

$\mathcal{Q}_{\mathcal{R}}=\frac{\kappa_{\mathcal{R}}}{(k+N)}\quad\text{and}\quad
c=\frac{N(Nk+1)}{N+k}.$ (62)

Next we want to calculate the ratio of modular transformation matrices
$\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}}{\mathcal{S}_{0\mathcal{R}}}$
appearing in equation (55). Instead of representing a Young diagram by the set
of box numbers $\\{n_{i}^{(a)}\\}$ we use the variables $\theta_{i}$ defined
in equation (34). Since in the double scaling limit $\theta_{i}$ ranges
between $-\pi$ and $\pi$, we can arrange them as eigenvalues of an $N\times N$
unitary matrix

$U=\text{diag}(e^{i\theta_{1}},\cdots,e^{i\theta_{N}}).$ (63)

After some calculation one can show that the ratio
$\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}}{\mathcal{S}_{0\mathcal{R}}}$
can be expressed as

$\displaystyle\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}}{\mathcal{S}_{0\mathcal{R}}}=\mathbf{s}_{\mathcal{R}_{a}}(U),$
(64)

where the Schur polynomial $\mathbf{s}_{\mathcal{R}_{a}}(U)$ of the unitary
group element $U$ in representation $\mathcal{R}_{a}$ is

$\displaystyle\mathbf{s}_{\mathcal{R}_{a}}(U)$
$\displaystyle=\frac{\textstyle\det\left[e^{i\theta_{i}h_{j}^{(a)}}\right]}{\det\left[e^{i\theta_{i}(N-j)}\right]}.$
(65)

Note that the notation $U$ used in equation (64) corresponds to representation
$\mathcal{R}$ on its left hand side i.e. the diagonal elements of $U$
correspond to $h_{i}$ (35) of $\mathcal{R}$. The unnormalised Hopf link
invariant, therefore can be expressed as111Note the difference between the
expressions of Hopf link here and in chakraborty2022large .

$\widetilde{{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},k)=q^{\frac{pN(Nk+1)}{24}}\sum_{\mathcal{R}}q^{-{p}\kappa_{\mathcal{R}}}~{}\mathcal{S}_{0\mathcal{R}}^{2}\
\mathbf{s}_{\mathcal{R}_{1}}(U)\mathbf{s}_{\mathcal{R}_{2}}(U),$ (66)

where $q$ is the $(k+N)$th root of unity (11). In equation (66) each
$\mathcal{R}$ corresponds to a unitary matrix $U$ in an ensemble of unitary
matrices. The summation over $\mathcal{R}$ therefore corresponds to summation
over unitary matrices. The factor $\mathcal{S}_{0\mathcal{R}}^{2}$ when
written in terms of $\theta_{i}$ variables, reduces to Vandermonde measure for
unitary matrices. Thus in the large $N$ limit we can write

$\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2}=\int\prod_{i}d\theta_{i}\prod_{i<j}\sin^{2}\left(\frac{\theta_{i}-\theta_{j}}{2}\right)\equiv\int[DU]$
(67)

where we have ignored few $\theta$ independent overall factors which doesn’t
affect our analysis. Hence the partition function (56) can be written as a
$0$-dimensional unitary matrix model with weight factor
$e^{-(N+k)^{2}S[\theta]}$ as

$\displaystyle\mathcal{Z}(S^{3}/\mathbb{Z}_{p},k)=\int\prod_{i}d\theta_{i}\prod_{i<j}\sin^{2}\left(\frac{\theta_{i}-\theta_{j}}{2}\right)e^{-(N+k)^{2}S[\theta]},$
(68)

where

$S[\theta]=\frac{p\lambda}{N\pi}\sum_{i=1}^{N}\left(\frac{\theta_{i}^{2}}{4}-\frac{\pi^{2}}{12}\right)+\frac{\pi
p\lambda(1-\lambda)}{12}.$ (69)

Note that, in order to arrive at the above result we have replaced $p$ in
equation (56) by $-ip$. The normalised Hopf link invariant (57) is therefore
given as the integral of two Schur polynomials
$\mathbf{s}_{\mathcal{R}_{1}}(U)$ and $\mathbf{s}_{\mathcal{R}_{2}}(U)$ over
the unitary ensemble with the aforementioned weight factor

$\displaystyle{{\mathscr{V}}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},k)$
$\displaystyle=\frac{1}{\mathcal{Z}(S^{3}/\mathbb{Z}_{p},k)}\int[DU]e^{-(N+k)^{2}S[\theta]}\mathbf{s}_{\mathcal{R}_{1}}(U)\mathbf{s}_{\mathcal{R}_{2}}(U)$
$\displaystyle=\left\langle\mathbf{s}_{\mathcal{R}_{1}}(U)\mathbf{s}_{\mathcal{R}_{2}}(U)\right\rangle.$
(70)

For large $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ (i.e. when the number of
boxes $n_{i}^{(1)}$ and $n_{i}^{(2)}$ become of order $N$ or $k$, in the
double scaling limit) it is difficult to calculate the correlation function
even using the saddle point approximation. However, if $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ are small, one can find the leading order contribution to
${\mathscr{V}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},k)$
by computing $\mathbf{s}_{\mathcal{R}_{1}}(U)$ and
$\mathbf{s}_{\mathcal{R}_{2}}(U)$ on the classical solution that extremizes
the partition function (56).

###  5.1 Leading contribution to $(L,L)$ torus link invariant for small
representations

For a (2,2) torus link invariant when the “size” of $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ placed on the component knots are much smaller than $N$,
they do not back react on the classical solution extremizing the partition
function. Hence we can consider the operators
$\mathbf{s}_{\mathcal{R}_{1}}(U)$ and $~{}\mathbf{s}_{\mathcal{R}_{2}}(U)$ as
probe and evaluate them on the background solution (72).

Let us define a distribution function for the variables $\\{\theta_{i}\\}$ in
the large $N$ limit as

$\rho(\theta)=\frac{1}{N}\sum_{i=1}^{N}\delta(\theta-\theta_{i}),\quad\text{such
that}\quad\int_{-\pi}^{\pi}d\theta\rho(\theta)=1.$ (71)

It was shown in Ref. Chakraborty:2021oyq  that in the double scaling limit (1)
the partition function (56) is dominated by the following distribution
function222In this paper we have used the no-cap solution. In the double
scaling limit CS theory admits a phase transition at some critical value of
$\lambda$. Beyond that critical value the dominant distribution is given by a
capped eigenvalue density. As a results the behaviour of correlation functions
and hence knot/link invariants will change. We have commented on this in the
conclusion section.

$\displaystyle\rho(\theta)$
$\displaystyle=\frac{p}{2\pi^{2}\lambda}\tanh^{-1}\sqrt{1-\frac{e^{-2\pi\lambda/p}}{\cos^{2}(\theta/2)}},\quad\text{with}~{}-2\sec^{-1}e^{\pi\lambda/p}<\theta<2\sec^{-1}e^{\pi\lambda/p}$
(72)

which is a solution to the following saddle point equation obtained from the
partition function (68)

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\rho(\theta^{\prime})\cot\left(\frac{\theta-\theta^{\prime}}{2}\right)d\theta^{\prime}=\frac{p}{2\pi\lambda}\theta.$
(73)

Therefore in the probe approximation the Hopf link invariant (70) can be
computed by evaluating the Schur polynomials on the solution (72)

${\mathscr{V}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},\lambda)=\mathbf{s}_{\mathcal{R}_{1}}(\bar{U})\mathbf{s}_{\mathcal{R}_{2}}(\bar{U})$
(74)

where $\bar{U}$ corresponds to the distribution (72).

The Schur polynomial admits a character expansion marino2005chern

$\displaystyle\mathbf{s}_{\mathcal{R}_{a}}(U)=\sum_{\vec{k}^{(a)}}\frac{\chi_{\mathcal{R}_{a}}(c(\vec{k}^{(a)}))}{z_{\vec{k}^{(a)}}}\Upsilon_{\vec{k}^{(a)}}(U),$
(75)

where

$\displaystyle\vec{k}^{(a)}$
$\displaystyle=\left(k^{(a)}_{1},k^{(a)}_{2},k^{(a)}_{3},...\right)~{}\text{such
that}~{}\sum_{r}rk^{(a)}_{r}=n^{(a)},$ (76)

and

$\displaystyle\Upsilon_{\vec{k}^{(a)}}(U)=\prod_{r}\left(\text{Tr}~{}U^{r}\right)^{k^{(a)}_{r}},\quad
z_{\vec{k}^{(a)}}=\prod_{r}k^{(a)}_{r}!~{}r^{k^{(a)}_{r}}.$ (77)

$n^{(a)}$ being the total number of boxes in representation $\mathcal{R}_{a}$.
$c(\vec{k}^{(a)})\equiv 1^{k^{(a)}_{1}}2^{k^{(a)}_{2}}3^{k^{(a)}_{3}}\cdots$
is the conjugacy class of the symmetric group $\mathbb{G}_{{n^{(a)}}}$ of
degree $n^{(a)}$, consisting of $k^{(a)}_{1}$ cycles of length 1,
$k^{(a)}_{2}$ cycles of length 2 and so on. In other words, $k^{(a)}_{r}$
denotes the number of $r$-cycle of the conjugacy class $c(\vec{k}^{(a)})$.
$\chi_{\mathcal{R}_{a}}(c(\vec{k}^{(a)}))$ is the corresponding character, in
representation $\mathcal{R}_{a}$.

Since to each vector $\vec{k}^{(a)}$ we can associate a conjugacy class
$c(\vec{k}^{(a)})$, the sum over $\vec{k}^{(a)}$ in (75) represents a sum over
all possible conjugacy classes subjected to the condition (76). Thus in the
probe approximation, computation of the Hopf link invariant reduces to
computation of the quantities $\Upsilon_{\vec{k}^{(a)}}(U)$ on the solution
(72).

In the large $N$ limit the expectation value of $\Tr U^{r}$ is dominated by
the background solution i.e

$\langle\Tr U^{r}\rangle\rightarrow{\Tr}\bar{U}^{r}=N\int
e^{ir\theta}\rho(\theta)d\theta=N\int\cos r\theta~{}\rho(\theta)d\theta\equiv
N\rho_{r},$ (78)

where

$\displaystyle\rho_{r}$ $\displaystyle=\int\cos
r\theta~{}\rho(\theta)d\theta,$ (79)

with $\rho(\theta)$ as given in (72). Therefore, in the large $N$ limit
$\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$ is given by

$\displaystyle\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$
$\displaystyle=\sum_{\vec{k}^{(a)}}\frac{\chi_{\mathcal{R}_{a}}(c(\vec{k}^{(a)}))}{z_{\vec{k}^{(a)}}}\prod_{r}N^{k^{(a)}_{r}}\rho_{r}^{k^{(a)}_{r}}=\sum_{\vec{k}^{(a)}}N^{K^{(a)}}\frac{\chi_{\mathcal{R}_{a}}(c(\vec{k}^{(a)}))}{z_{\vec{k}^{(a)}}}\prod_{r}\rho_{r}^{k^{(a)}_{r}},$
(80) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{where}~{}\quad
K^{(a)}=\sum_{r}k^{(a)}_{r}.$

For small representations $k^{(a)}_{r}$’s are $\mathcal{O}(1)$ numbers. Hence
$z_{\vec{k}^{(a)}}$’s are also $\mathcal{O}(1)$ numbers. As a result the $N$
dependence of $\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$ is given by
$N^{K^{(a)}}$. Therefore, in the large $N$ limit
$\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$ is dominated by a conjugacy class for
which $K^{(a)}$ is maximum subjected to (76). Using (76) we can replace
$k^{(a)}_{1}$ in $K^{(a)}$ and find

$K^{(a)}=n^{(a)}-k^{(a)}_{2}-2k^{(a)}_{3}-\cdots.$ (81)

Hence, the maximum value of $K^{(a)}$ is given by $k^{(a)}_{1}=n^{(a)}$ and
$k^{(a)}_{2}=k^{(a)}_{3}=\cdots=0$. This choice reduces equation (76) to
$\vec{k}^{(a)}=\left(n^{(a)},0,0,...\right)$, as a result the Schur polynomial
(80) evaluated on the classical solution becomes

$\displaystyle\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$
$\displaystyle=N^{n^{(a)}}\rho_{1}^{n^{(a)}}\frac{\chi_{\mathcal{R}_{a}}(1^{n^{(a)}})}{(n^{(a)})!},$
(82)

where $\rho_{1}$ can be evaluated using (79) and is given by

$\rho_{1}=p\left(\frac{1-e^{-\frac{2\pi\lambda}{p}}}{2\pi\lambda}\right),$
(83)

$\chi_{\mathcal{R}_{a}}\left(1^{n^{(a)}}\right)$ is the character of the
conjugacy class consisting of $n^{(a)}$ number of 1-cycles in representation
$\mathcal{R}_{a}$. Thus the leading contribution (denoted by
$\bm{\ell}.\bm{c}.$) to the unknot invariant in fundamental representation is
given by

$\bm{\ell}.\bm{c}.[{\mathscr{V}}_{\Box}^{\left(2,1\right)}\left(S^{3}/\mathbb{Z}_{p},\lambda\right)]=\mathbf{s}_{\Box}(\bar{U})=N\rho_{1},$
(84)

The superscript $(2,1)$ refers to an unknot obtained as closure of a two
strand braid with one crossing. Therefore, the leading contribution to the
Hopf link invariant in the double scaling limit can be expressed as (74)

$\displaystyle\bm{\ell}.\bm{c}.\left[{\mathscr{V}}^{(2,2)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},\lambda)\right]$
$\displaystyle=N^{n^{(1)}+n^{(2)}}\rho_{1}^{n^{(1)}+n^{(2)}}\frac{\chi_{\mathcal{R}_{1}}\left(1^{n^{(1)}}\right)\chi_{\mathcal{R}_{2}}\left(1^{n^{(2)}}\right)}{\left(n^{(1)}\right)!\left(n^{(2)}\right)!}$
$\displaystyle=\bigg{(}\bm{\ell}.\bm{c}.$
$\displaystyle\left[{\mathscr{V}}_{\Box}^{\left(2,1\right)}\left(S^{3}/\mathbb{Z}_{p},\lambda\bigg{)}\right]\right)^{n^{(1)}+n^{(2)}}\frac{\chi_{\mathcal{R}_{1}}\left(1^{n^{(1)}}\right)\chi_{\mathcal{R}_{2}}\left(1^{n^{(2)}}\right)}{\left(n^{(1)}\right)!\left(n^{(2)}\right)!}$
(85)

It shows that the leading contribution of the Hopf link invariant with small
representation placed on the component knots is proportional to appropriate
powers of the leading contribution of unknot invariant in fundamental
representation. Likewise, the leading contribution of unknot carrying
arbitrary representation is

$\displaystyle\bm{\ell}.\bm{c}.\left[{\mathscr{V}}_{\mathcal{R}_{1}}^{\left(2,1\right)}(S^{3}/\mathbb{Z}_{p},\lambda)\right]$
$\displaystyle=\left(\bm{\ell}.\bm{c}.\left[{\mathscr{V}}_{\Box}^{\left(2,1\right)}(S^{3}/\mathbb{Z}_{p},\lambda)\right]\right)^{n^{(1)}}\frac{\chi_{\mathcal{R}_{1}}\left(1^{n^{(1)}}\right)}{\left(n^{(1)}\right)!}.$
(86)

We can generalise (85) for $(L,L)$ torus link invariant in the probe
approximation and it is given by

$\displaystyle\bm{\ell}.\bm{c}.\left[{\mathscr{V}}^{(L,L)}_{\mathcal{R}_{1},\cdots\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},\lambda)\right]=\prod_{a=1}^{L}N^{n^{(a)}}\rho_{1}^{n^{(a)}}~{}\frac{\chi_{\mathcal{R}_{a}}\left(1^{n^{(a)}}\right)}{\left(n^{(a)}\right)!}\equiv\prod_{a=1}^{L}\bm{\ell}.\bm{c}.\left[{\mathscr{V}}_{\mathcal{R}_{a}}^{\left(2,1\right)}(S^{3}/\mathbb{Z}_{p},\lambda)\right].$
(87)

For $(3,3)$ torus link, with the three component knots carrying some specific
representations with small number of boxes in the corresponding Young
diagrams, we tabulate the colored HOMFLY-PT polynomials and the leading
contributions in Table LABEL:Tableof3,3lc. Note that in order to match the
matrix model result (87) with the tabulated ones we need to follow the
prescription as mentioned below:

  * $\bullet$

first, replace $p\rightarrow ip$ in the expressions obtained by matrix model
calculations to compensate the replacement $p\rightarrow-ip$ done earlier (see
below equation (69)),

  * $\bullet$

then set $p=1$, since the colored HOMFLY-PT polynomials are obtained for
framed links embedded inside $S^{3}$.

Explicitly, it implies that

$\displaystyle\mathscr{V}^{(L,L)}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},\lambda)\overset{\begin{subarray}{c}p\rightarrow
ip,\\\
p=1\end{subarray}}{\longrightarrow}\mathcal{W}^{[(L,L),f=\\{1,...,1\\}]}_{\mathcal{R}_{1},...,\mathcal{R}_{L}}(S^{3},\lambda).$
(88)

When these modifications are taken into account, the expression of $\rho_{1}$
(83) becomes identical to the leading order contribution of the vertical
framing $U(N)$ invariant $\mathcal{W}_{\Box}^{(2,1)}(S^{3},\lambda)$ of an
unknot with fundamental representation placed on its component.

The leading contributions as obtained from the colored HOMFLY-PT polynomials
of $(3,3)$ torus links (refer to Table LABEL:Tableof3,3lc) are in exact
agreement with those (87) obtained using the eigenvalue density (72)
extremising the partition function. It is to be noted that, in the large $N$
limit the evaluation of leading contribution using the eigenvalue density of
the dominant representation is only possible when the representations placed
on the component knots of the link are much smaller as compared to the rank
$N$ of the gauge group at consideration.

Table 1: $(3,3)$ Torus link invariants and their leading contributions $\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2},\mathcal{R}_{3}}^{(3,3)}(q,v)$ |  The colored HOMFLY-PT polynomial |  Leading contribution
---|---|---
$\mathcal{W}^{(3,3)}_{\ydiagram[]{1},\ydiagram[]{1},\ydiagram[]{1}}$ |  $(-1+q)^{-3}q^{-3/2}\big{(}-1+2q-2q^{2}+q^{3}-2q^{4}+2q^{5}-q^{6}+v-qv+q^{2}v+$ $q^{3}v+q^{4}v-q^{5}v+q^{6}v-qv^{2}+q^{2}v^{2}-3q^{3}v^{2}+q^{4}v^{2}-q^{5}v^{2}+q^{3}v^{3}\big{)}$ |  $\frac{i(-1+e^{2\pi i\lambda})^{3}N^{3}}{8\pi^{3}\lambda^{3}}$
$\mathcal{W}_{\ydiagram[]{2},\ydiagram[]{1},\ydiagram[]{1}}^{(3,3)}$ |  $(-1+q)^{-4}q^{-2}(1+q)^{-1}\big{(}1-2q+q^{2}+q^{3}-q^{4}+q^{5}-q^{7}+q^{8}-v+qv-q^{3}v-q^{4}v-q^{6}v-q^{9}v+qv^{2}-q^{2}v^{2}+q^{3}v^{2}+$ $q^{4}v^{2}+q^{5}v^{2}+q^{6}v^{2}+q^{7}v^{2}+q^{9}v^{2}-q^{3}v^{3}+q^{4}v^{3}-2q^{5}v^{3}-q^{6}v^{3}-q^{8}v^{3}+q^{6}v^{4}\big{)}$ |  $\frac{(-1+e^{2\pi i\lambda})^{4}N^{4}}{32\pi^{4}\lambda^{4}}$
$\mathcal{W}_{\ydiagram[]{2},\ydiagram[]{2},\ydiagram[]{1}}^{(3,3)}$ |  $(-1+q)^{-5}q^{-7/2}(1+q)^{-2}\big{(}-1+2q-3q^{3}+2q^{4}+q^{5}-3q^{6}+2q^{7}+q^{8}-3q^{9}+q^{10}+q^{11}-q^{12}+v-qv-q^{2}v+2q^{3}v+q^{6}v-q^{7}v+q^{8}v+2q^{9}v-q^{11}v+q^{12}v+q^{13}v-qv^{2}$ $+q^{2}v^{2}-q^{4}v^{2}-q^{5}v^{2}-q^{7}v^{2}-q^{8}v^{2}-2q^{9}v^{2}-2q^{10}v^{2}+q^{11}v^{2}-q^{12}v^{2}-2q^{13}v^{2}+q^{3}v^{3}-q^{4}v^{3}+3q^{6}v^{3}-q^{7}v^{3}+5q^{9}v^{3}+2q^{12}v^{3}+q^{13}v^{3}-q^{6}v^{4}+q^{7}v^{4}-q^{8}v^{4}-2q^{9}v^{4}-q^{11}v^{4}-q^{12}v^{4}+q^{10}v^{5}\big{)}$ |  $-\frac{i(-1+e^{2\pi i\lambda})^{5}N^{5}}{128\pi^{5}\lambda^{5}}$
$\mathcal{W}_{\ydiagram[]{2},\ydiagram[]{2},\ydiagram[]{2}}^{(3,3)}$ |  $(-1+q)^{6}q^{-6}(1+q)^{-3}\big{(}1-2q-q^{2}+5q^{3}-2q^{4}-4q^{5}+4q^{6}+q^{7}-3q^{8}+2q^{10}+q^{12}-2q^{13}-2q^{14}+4q^{15}-2q^{17}+q^{18}-v+qv+2q^{2}v-3q^{3}v-q^{4}v+3q^{5}v-2q^{6}v-q^{7}v+3q^{8}v-2q^{9}v-2q^{10}v+q^{11}v-3q^{12}v+q^{13}v+4q^{14}v-4q^{15}v-3q^{16}v+2q^{17}v-q^{19}v+qv^{2}-q^{2}v^{2}-$ $q^{3}v^{2}+2q^{4}v^{2}+q^{7}v^{2}-q^{8}v^{2}+q^{9}v^{2}+2q^{10}v^{2}+2q^{11}v^{2}+q^{12}v^{2}+2q^{13}v^{2}-q^{14}v^{2}+5q^{16}v^{2}+q^{17}v^{2}-q^{18}v^{2}+2q^{19}v^{2}-q^{3}v^{3}+q^{4}v^{3}+q^{5}v^{3}-3q^{6}v^{3}+2q^{8}v^{3}-3q^{9}v^{3}-q^{10}v^{3}-5q^{12}v^{3}-$ $2q^{13}v^{3}+q^{14}v^{3}-3q^{15}v^{3}-3q^{16}v^{3}-q^{17}v^{3}-2q^{18}v^{3}-q^{19}v^{3}+q^{6}v^{4}-q^{7}v^{4}-q^{8}v^{4}+4q^{9}v^{4}-4q^{11}v^{4}+6q^{12}v^{4}+4q^{13}v^{4}-2q^{14}v^{4}+2q^{15}v^{4}+3q^{16}v^{4}+q^{17}v^{4}+2q^{18}v^{4}-q^{10}v^{5}+q^{11}v^{5}-3q^{13}v^{5}-2q^{16}v^{5}-q^{17}v^{5}+q^{15}v^{6}\big{)}$ |  $-\frac{(-1+e^{2\pi i\lambda})^{6}N^{6}}{512\pi^{6}\lambda^{6}}$
$\mathcal{W}_{\ydiagram[]{2,1},\ydiagram[]{1},\ydiagram[]{1}}^{(3,3)}$ |  $(-1+q)^{-5}q^{-5/2}(1+q+q^{2})^{-1}\big{(}-q+2q^{2}-2q^{3}+2q^{4}-4q^{5}+5q^{6}-4q^{7}+2q^{8}-2q^{9}+2q^{10}-q^{11}+v-qv+q^{2}v-q^{3}v+3q^{4}v-q^{5}v+q^{6}v-q^{7}v+3q^{8}v-q^{9}v+q^{10}v-q^{11}v+q^{12}v-v^{2}+qv^{2}-2q^{2}v^{2}+2q^{3}v^{2}-6q^{4}v^{2}+4q^{5}v^{2}-6q^{6}v^{2}+$ $4q^{7}v^{2}-6q^{8}v^{2}+2q^{9}v^{2}-2q^{10}v^{2}+q^{11}v^{2}-q^{12}v^{2}+qv^{3}-q^{2}v^{3}+2q^{3}v^{3}+4q^{5}v^{3}-2q^{6}v^{3}+4q^{7}v^{3}+2q^{9}v^{3}-q^{10}v^{3}+q^{11}v^{3}-q^{3}v^{4}+q^{4}v^{4}-3q^{5}v^{4}+q^{6}v^{4}-3q^{7}v^{4}+q^{8}v^{4}-q^{9}v^{4}+q^{6}v^{5}\big{)}$ |  $-\frac{i(-1+e^{2\pi i\lambda})^{5}N^{5}}{96\pi^{5}\lambda^{5}}$
$\mathcal{W}_{\ydiagram[]{2,1},\ydiagram[]{2},\ydiagram[]{1}}^{(3,3)}$ |  $(-1+q)^{6}q^{-4}(1+q)^{-1}(1+q+q^{2})^{-1}\big{(}q-2q^{2}+q^{3}+q^{5}-2q^{7}+2q^{8}-q^{9}+q^{10}-q^{12}+2q^{13}-2q^{14}+q^{15}-v+qv-q^{4}v-q^{5}v-2q^{10}v-q^{11}v-q^{16}v+v^{2}-qv^{2}+q^{2}v^{2}-q^{3}v^{2}+2q^{4}v^{2}+q^{5}v^{2}+q^{6}v^{2}+q^{7}v^{2}+2q^{9}v^{2}+2q^{10}v^{2}+2q^{11}v^{2}+2q^{12}v^{2}-q^{13}v^{2}+$ $2q^{14}v^{2}-q^{15}v^{2}+2q^{16}v^{2}-qv^{3}+q^{2}v^{3}-q^{3}v^{3}-2q^{5}v^{3}-q^{6}v^{3}-2q^{7}v^{3}-q^{8}v^{3}-2q^{9}v^{3}-3q^{10}v^{3}-3q^{11}v^{3}-q^{12}v^{3}-2q^{13}v^{3}-q^{15}v^{3}-q^{16}v^{3}+q^{3}v^{4}-q^{4}v^{4}+q^{5}v^{4}+q^{6}v^{4}+2q^{7}v^{4}+q^{8}v^{4}+q^{9}v^{4}+3q^{10}v^{4}+3q^{11}v^{4}+2q^{13}v^{4}+q^{15}v^{4}-q^{6}v^{5}+q^{7}v^{5}-2q^{8}v^{5}-2q^{10}v^{5}-q^{11}v^{5}-q^{13}v^{5}+q^{10}v^{6}\big{)}$ |  $-\frac{(-1+e^{2\pi i\lambda})^{6}N^{6}}{384\pi^{6}\lambda^{6}}$

For a large representation $\mathcal{R}_{a}$, the total number of boxes
$n^{(a)}$ is of $\mathcal{O}(N)$ which implies some of the $k^{(a)}_{r}$’s
(76) can also become of $\mathcal{O}(N)$. As a result the denominator
$z_{\vec{k}}$ in equation (75) will no longer be of $\mathcal{O}(1)$. Hence
the above saddle point analysis will not give correct answer in such
scenarios. However, we have a way to find the leading contribution for a class
of large representations which will be addressed in the following subsection.

###  5.2 Leading contribution to $(L,L)$ torus link invariant for large
symmetric representations

In this section, we provide a large $N$ expression for $(L,L)$ link invariant
when all the representations are large symmetric representations. The $U(N)$
invariant for $(L,L)$ torus link, embedded inside three manifold
$S^{3}/\mathbb{Z}_{p}$, carrying integrable representations
$\mathcal{R}_{1},...,\mathcal{R}_{L}$ on the component knots is given by

${\mathscr{V}}^{(L,L)}_{\mathcal{R}_{1},\cdots,\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},k)=\frac{1}{\mathcal{Z}(S^{3}/\mathbb{Z}_{p},k)}\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}\prod_{a=1}^{L}\mathbf{s}_{\mathcal{R}_{a}}(U)\equiv\left\langle\prod_{a=1}^{L}\mathbf{s}_{\mathcal{R}_{a}}(U)\right\rangle.$
(89)

Using the character expansion for Schur polynomial (75) and (67), in the
double scaling limit we can express the above invariant as

$\mathscr{V}^{(L,L)}_{\mathcal{R}_{1},\cdots\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},\lambda)=\frac{1}{\mathcal{Z}(S^{3}/\mathbb{Z}_{p},\lambda)}\sum_{\vec{k}^{(1)},\cdots\vec{k}^{(L)}}\prod_{a=1}^{L}\frac{\chi_{\mathcal{R}_{a}}(c(\vec{k}^{(a)}))}{z_{\vec{k}^{(a)}}}\int[D\theta]e^{-N^{2}S^{\left(\vec{k}^{(1)},\cdots,\vec{k}^{(L)}\right)}_{eff}[\rho(\theta)]}$
(90)

where

$\displaystyle S^{(\vec{k}^{(1)},\cdots,\vec{k}^{(L)})}_{eff}[\rho(\theta)]$
$\displaystyle=-\frac{1}{2}\int
d\theta\rho(\theta)\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
d\theta^{\prime}\rho(\theta^{\prime})\log\left[4\sin^{2}\left(\frac{\theta-\theta^{\prime}}{2}\right)\right]$
$\displaystyle+\frac{p}{\pi\lambda}\int\rho(\theta)\left(\frac{\theta^{2}}{4}-\frac{\pi^{2}}{12}\right)d\theta+\frac{\pi
p(1-\lambda)}{12\lambda}$
$\displaystyle-\frac{1}{N^{2}}\sum_{r}(k_{r}^{(1)}+\cdots+k_{r}^{(L)})\ln\left[N\int
d\theta\rho(\theta)e^{ir\theta}\right].$ (91)

For a given $\vec{k}^{(1)},\cdots,\vec{k}^{(L)}$,
$\int[D\theta]e^{-N^{2}S^{(\vec{k}^{(1)},\cdots,\vec{k}^{(L)})}_{eff}[\rho(\theta)]}$
is dominated by a classical configuration satisfying the saddle point equation

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\rho(\theta^{\prime})\cot\left(\frac{\theta-\theta^{\prime}}{2}\right)d\theta^{\prime}=\frac{p}{2\pi\lambda}\theta-\frac{i}{N^{2}}\sum_{r}\frac{r(k_{r}^{(1)}+\cdots+k_{r}^{(L)})}{\rho_{r}}e^{ir\theta}.$
(92)

This is a coupled equation because the moments of the distribution
$\rho_{r}=\int d\theta\rho(\theta)e^{ir\theta}$ appear in the last term and
hence difficult to solve. The last term is proportional to $1/N^{2}$.
Apparently, it seems that this term will not be important to find
$\rho(\theta)$ in the large $N$ limit. But if any of the $\mathcal{R}_{a}$ is
large such that $n^{(a)}$ is of $\mathcal{O}(N^{2})$, then following (76) some
of the $k^{(a)}_{r}$ can also be of $\mathcal{O}(N^{2})$. In such cases we can
not neglect the last term to solve for $\rho(\theta)$.

However, in the case of all representations ($\mathcal{R}_{a}$) large but
completely symmetric, there can be at most $k$ boxes in the first row and
hence $n^{(a)}\leq k$. As a result all the $k_{r}^{(a)}$s are less than (or
equal to) $k$. Therefore in the large $N$ limit we can still neglect the last
term. This suggest us that in order to compute
$\mathscr{V}^{(L,L)}_{\mathcal{R}_{1},\cdots\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},\lambda)$
for large symmetric representations we can still use the probe approximation
and calculate the Schur polynomials on the background solution (72). Thus the
$(L,L)$ link invariant (89) becomes

$\mathscr{V}^{(L,L)}_{\mathcal{R}_{1},\cdots\mathcal{R}_{L}}(S^{3}/\mathbb{Z}_{p},\lambda)=\prod_{a=1}^{L}\mathbf{s}_{\mathcal{R}_{a}}(\bar{U}).$
(93)

where $\bar{U}$ corresponds to the density distribution solving (73). Note
that, unlike (82) $\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$ will no longer be
dominated by the conjugacy class having $n^{(a)}$ number of 1-cycles.

Since the character of any conjugacy class of the permutation group for a
completely symmetric representation is 1, the Schur polynomial (80) can be
written as,

$\displaystyle\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})$
$\displaystyle=\sum_{\vec{k}^{(a)}}\prod_{r}\frac{(N\rho_{r})^{k^{(a)}_{r}}}{k^{(a)}_{r}!~{}r^{k^{(a)}_{r}}}$
$\displaystyle=\sum_{\vec{k}^{(a)}}\exp\left[\sum_{r}\left(k^{(a)}_{r}\ln\left(\frac{N\rho_{r}}{r}\right)-k^{(a)}_{r}\ln
k^{(a)}_{r}+k^{(a)}_{r}\right)\right]$ (94)

subjected to the condition ${\sum_{r}}rk^{(a)}_{r}=n^{(a)}$. We use the
Lagrange multiplier $\alpha^{(a)}$ to extremise the above summation. Suppose

$\mathcal{H}^{(a)}=\sum_{r}\left(k^{(a)}_{r}\ln\left(\frac{N\rho_{r}}{r}\right)-k^{(a)}_{r}\ln
k^{(a)}_{r}+k^{(a)}_{r}\right)+\alpha^{(a)}\left(\sum_{r}rk^{(a)}_{r}-n^{(a)}\right).$
(95)

Extremising $\mathcal{H}^{(a)}$ with respect to $k^{(a)}_{r}$ we find,

$k^{(a)}_{r}=\frac{N\rho_{r}}{r}e^{r\alpha^{(a)}}.$ (96)

The Lagrange multiplier $\alpha^{(a)}$ can be obtained by substituting
$k^{(a)}_{r}$ in the equation $\sum_{r}rk^{(a)}_{r}=n^{(a)}$

$\sum_{r}\rho_{r}e^{r\alpha^{(a)}}=\frac{n^{(a)}}{N}.$ (97)

Finally evaluating $\mathcal{H}^{(a)}$ on the solution we find

$\overline{\mathcal{H}}^{(a)}=\sum_{r}\frac{N\rho_{r}}{r}e^{r\alpha^{(a)}}-\alpha^{(a)}~{}n^{(a)}.$
(98)

Hence we have

$\mathbf{s}_{\mathcal{R}_{a}}(\bar{U})=e^{\overline{\mathcal{H}}^{(a)}}.$ (99)

We can use this result to find the Schur polynomial evaluated on the classical
solution. Analytically it is challenging to find the Schur polynomial for any
arbitrary values of $\lambda$. However, we can find it for small $\lambda$.

For small $\lambda$ the eigenvalue distribution (72) is given by

$\rho(\theta)=\frac{p}{2\pi^{2}\lambda}\sqrt{\frac{2\pi\lambda}{p}-\frac{\theta^{2}}{4}}.$
(100)

We can compute $\rho_{r}$ by using the formula (79) with $\rho(\theta)$ as
defined above and range of $\theta$ lying from $-2\sqrt{{2\pi\lambda}/{p}}$ to
$2\sqrt{{2\pi\lambda}/{p}}$. The result is as follows

$\rho_{r}=1-\frac{r^{2}\pi\lambda}{p}.$ (101)

Using (97) we can find that the Lagrange multiplier is given by,

$\displaystyle\alpha^{(a)}=\ln\left(\frac{\kappa^{(a)}}{1+\kappa^{(a)}}\right)+\frac{(1+2\kappa^{(a)})\pi\lambda}{p}$
(102)

where $\kappa^{(a)}=\frac{n^{(a)}}{N}$. Hence $\overline{\mathcal{H}}^{(a)}$
can be evaluated to obtain,

$\overline{\mathcal{H}}^{(a)}=\ln\mathbf{s}_{\mathcal{R}_{a}}=(1+\kappa^{(a)})\ln(1+\kappa^{(a)})-\kappa^{(a)}\ln\kappa^{(a)}-\frac{\kappa^{(a)}(1+\kappa^{(a)})\pi\lambda}{p}.$
(103)

Substituting $\overline{\mathcal{H}}^{(a)}$ in equation (99) and then using
(93) we can deduce the leading order contribution for $(L,L)$ torus link
invariant whose components carry arbitrary large symmetric representations.

To understand how different components are intertwined in a link we have to go
beyond the leading contribution. From the matrix model point of view, the
subleading correction is captured by the connected piece of the correlator.
The computation of the connected piece becomes tedious involving resolvent and
contour integration for a general $(L,L)$ torus link. However, one can extract
the subleading contributions from the colored HOMFLY-PT polynomials (as in
Table LABEL:Tableof3,3lc) when small representations are placed on the
component knots of the torus link.

In order to confirm that the subleading contribution from the colored HOMFLY-
PT matches with the matrix model approach, we will explicitly present the
computation of connected correlator for the Hopf link invariant in the
following section.

##  6 Subleading corrections to Hopf link invariant

In this section we compute the subleading contribution to the invariant of
Hopf link. From (70) we note that the Hopf link invariant is equal to
correlation of two Schur polynomials. Let us consider that both the component
knots of the link are in fundamental representation i.e. ${\cal R}_{1}={\cal
R}_{2}=\Box$. The character expansion of the Schur polynomial (75) tells us
that $\mathbf{s}_{\Box}(U)=\Tr U$. Thus the Hopf link invariant with both the
components in fundamental representations can be expressed as

$\displaystyle\mathscr{V}^{(2,2)}_{\Box,\Box}(S^{3}/\mathbb{Z}_{p},k)=\langle\Tr
U\Tr U\rangle.$ (104)

Likewise, an unknot invariant in fundamental representation is given by

$\displaystyle\mathscr{V}^{(2,1)}_{\Box}(S^{3}/\mathbb{Z}_{p},k)=\langle\Tr
U\rangle.$ (105)

We introduce a rescaled trace operator

$\mathscr{T}\mathrm{r}U^{m}=\frac{1}{N}\Tr U^{m},\quad m\in\mathbb{Z}$ (106)

such that $\langle\mathscr{T}\mathrm{r}U^{m}\rangle$, when evaluated on saddle
point gives a $\mathcal{O}(1)$ number, i.e. the leading contribution to
$\langle\mathscr{T}\mathrm{r}U^{m}\rangle$ becomes of $\mathcal{O}(1)$. The
corrections to $\langle\mathscr{T}\mathrm{r}U^{m}\rangle$ are suppressed by
higher powers of $N$. In terms of rescaled trace operators, the Hopf link and
unknot invariants can be written as

$\mathscr{V}^{(2,2)}_{\Box,\Box}(S^{3}/\mathbb{Z}_{p},k)=N^{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle,\quad\mathscr{V}^{(2,1)}_{\Box}(S^{3}/\mathbb{Z}_{p},k)=N\langle\mathscr{T}\mathrm{r}U\rangle.$
(107)

We define the _connected piece_ of a two-point function in the following way
eynard2015random ,

$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\equiv
N^{2}\left(\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle-\langle\mathscr{T}\mathrm{r}U\rangle^{2}\right),$
(108)

where we have put an overall factor of $N^{2}$ on the right hand side such
that the leading contribution of
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$ is again of
$\mathcal{O}(1)$. This connected piece can be expressed in terms of the Hopf
link and unknot invariants as

$\mathscr{V}^{(2,2)}_{\Box,\Box}(S^{3}/\mathbb{Z}_{p},k)-\left(\mathscr{V}^{(2,1)}_{\Box}(S^{3}/\mathbb{Z}_{p},k)\right)^{2}=\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}.$
(109)

In the large $N$ limit $\langle\mathscr{T}\mathrm{r}U\rangle$ admits the
expansion

$\langle\mathscr{T}\mathrm{r}U\rangle=\langle\mathscr{T}\mathrm{r}U\rangle_{0}+\frac{1}{N^{2}}\langle\mathscr{T}\mathrm{r}U\rangle_{1}+\mathcal{O}\left(\frac{1}{N^{4}}\right)$
(110)

where $\langle\mathscr{T}\mathrm{r}U\rangle_{0}\equiv\rho_{1}$ (83) is the
expectation value of $\mathscr{T}\mathrm{r}U$ evaluated on the saddle point
(72) and $\langle\mathscr{T}\mathrm{r}U\rangle_{1}$ is the subleading
contribution. Therefore from (108) we see that the subleading contribution to
the two point correlator

$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle=\langle\mathscr{T}\mathrm{r}~{}U\rangle_{0}^{2}+\frac{1}{N^{2}}\bigg{(}2\langle\mathscr{T}\mathrm{r}U\rangle_{0}\langle\mathscr{T}\mathrm{r}U\rangle_{1}+\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\bigg{)}+\mathcal{O}\left(\frac{1}{N^{4}}\right).$
(111)

is the bracketed term multiplying $1/N^{2}$. It suggests that at the large $N$
perturbation, we can write the Hopf link and unknot invariants (107) as

$\displaystyle\mathscr{V}^{(2,2)}_{\Box,\Box}(S^{3}/\mathbb{Z}_{p},\lambda)$
$\displaystyle=N^{2}\langle\mathscr{T}\mathrm{r}U\rangle_{0}^{2}+\bigg{(}2\langle\mathscr{T}\mathrm{r}U\rangle_{0}\langle\mathscr{T}\mathrm{r}U\rangle_{1}+\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\bigg{)}+\mathcal{O}\left(\frac{1}{N^{2}}\right)$
(112) $\displaystyle\mathscr{V}^{(2,1)}_{\Box}(S^{3}/\mathbb{Z}_{p},\lambda)$
$\displaystyle=N\langle\mathscr{T}\mathrm{r}U\rangle_{0}+\frac{1}{N}\langle\mathscr{T}\mathrm{r}U\rangle_{1}+\mathcal{O}\left(\frac{1}{N^{3}}\right).$

The subleading correction to Hopf link invariant has two parts.
$\langle\mathscr{T}\mathrm{r}U\rangle_{0}\langle\mathscr{T}\mathrm{r}U\rangle_{1}$,
which being a product of the leading and subleading contributions to the
unknot invariant is insensitive to the intertwining between the two component
knots constituting Hopf link. The other contribution
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$, viz. the
connected piece of the two-point correlator, contains information about how
the two unknots are intertwined in a Hopf link when all the representations
are in fundamental. That is, it contains the information about the topological
nature of the Hopf link. Instead of fundamental, if we consider the Hopf link
in some other representations, the structure of the right hand side of (109)
will be very different. We shall discuss this at the end of this section.

Our next goal is to compute $\langle\mathscr{T}\mathrm{r}U\rangle_{1}$ and
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$ using the
methods developed in eynard2015random ; Ambjorn:1992gw 333The techniques to
compute subleading corrections developed in Ref. eynard2015random ;
Ambjorn:1992gw  are for hermitian matrix models. Here we develop the similar
techniques for unitary matrix models. and show that the matrix model results
are consistent to those obtained from the HOMFLY-PT polynomials.

###  6.1 Calculation of $\langle\mathscr{T}\mathrm{r}U\rangle_{1}$ and
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$

Introducing a set of complex variables $z_{i}=e^{i\theta_{i}}$, the partition
function (68) can be written as

$\mathcal{Z}(S^{3}/\mathbb{Z}_{p})=\int\prod_{i}dz_{i}\Delta(z_{i})e^{-N\sum_{i=1}^{N}V(z_{i})}$
(113)

where $\Delta(z_{i})=\prod_{i<j}(z_{i}-z_{j})^{2}$ and the potential
$V(z_{i})$ is given by

$V(z_{i})=-\frac{p}{4\pi\lambda}(\ln z_{i})^{2}+\ln z_{i}-\frac{\pi p}{12}.$
(114)

In the large $N$ limit with the continuous variables $x=i/N$,
$z_{i}\rightarrow z(x)$ and $\sum_{i}=N\int_{0}^{1}dx$, we can write

$\displaystyle-N\sum_{i=1}^{N}V(z_{i})=-N^{2}\int_{0}^{1}dxV(z(x))$ (115)

where,

$V(z)=-\frac{p}{4\pi\lambda}(\ln z)^{2}+\ln z-\frac{\pi p}{12}.$ (116)

Following (71) we define the eigenvalue density in the complex $z$-plane as

$\rho(z)=\left\langle\frac{1}{N}\sum_{i=1}^{N}\delta(z-z_{i})\right\rangle.$
(117)

In the large $N$ limit the above density satisfies the saddle point equation:

$2\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int dw\frac{\rho(w)}{z-w}=V^{\prime}(z).$ (118)

####  6.1.1 One-loop correlator

We introduce the one-loop correlator (also known as resolvent) $R(z)$ as

$R(z)=\frac{1}{N}\left\langle\Tr\frac{1}{z-U}\right\rangle\equiv\frac{1}{N}\sum_{m=0}^{\infty}z^{-1-m}\langle\Tr
U^{m}\rangle.$ (119)

Since $\langle\Tr U^{m}\rangle/N\leq 1$ for any $m$, $R(z)$ is an analytic
function in complex $z$ plane inside and outside the circle $|z|=1$. In
diagonal basis eigenvalues of $U$ lies on $|z|=1$ circle. Hence the function
$R(z)$ has poles at the eigenvalues of $U$. In the large $N$ limit, the
eigenvalues get accumulated and form a distribution on disconnected finite
arcs on the unit circle. As a result $R(z)$ develops branch cut singularities
on these arcs. Using (117) we can write

$R(z)=\int dw\frac{\rho(w)}{z-w}.$ (120)

Using the Sokhotski-Plemelj theorem

$\displaystyle\lim_{\epsilon\rightarrow 0}\int dx\frac{f(x)}{x_{0}-x\mp
i\epsilon}$ $\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int dx\frac{f(x)}{x_{0}-x}\pm i\pi f(x_{0})$ (121)

we find that the resolvent satisfies

$\lim_{\epsilon\rightarrow
0}\big{(}R(z+i\epsilon)+R(z-i\epsilon)\big{)}=V^{\prime}(z)$ (122)

and

$\lim_{\epsilon\rightarrow 0}\big{(}R(z+i\epsilon)-R(z-i\epsilon)\big{)}=-2\pi
i\rho(z).$ (123)

Moreover $R(z)$ has the following asymptotic $(z\rightarrow\infty)$ structure

$R(z\rightarrow\infty)=\frac{1}{z}+\frac{1}{z^{2}}\langle\mathscr{T}\mathrm{r}~{}U\rangle+\frac{1}{z^{3}}\langle\mathscr{T}\mathrm{r}~{}U^{2}\rangle+\mathcal{O}(1/z^{4}).$
(124)

Thus $R(z)$ is the generating function for
$\langle\mathscr{T}\mathrm{r}~{}U^{m}\rangle$ for different $m>0$. The
resolvent $R(z)$ admits a perturbative large $N$ expansion

$R(z)=R_{0}(z)+\frac{1}{N^{2}}R_{1}(z)+\cdots$ (125)

The leading contribution of the resolvent, $R_{0}(z)$ can be obtained by
solving the Dyson-Schwinger equation Friedan:1980tu ; Dutta:2016byx  in the
$N\rightarrow\infty$ limit. $R_{0}(z)$ also has a branch cut on the unit
circle. The eigenvalue density $\rho(z)$ can be determined from the
discontinuity of $R_{0}(z)$ on the branch cut. The higher genus contributions
$R_{1}(z)$, $R_{2}(z)$ et cetra then can be calculated using loop equations
order by order Ambjorn:1992gw . For a given potential $V(w)$, if the
eigenvalue density has support between $a$ and $b$ on the unit circle then
$R_{0}(z)$ can be expressed as Migdal:1983qrz

$R_{0}(z)=\frac{1}{2}\oint_{\mathcal{C}}\frac{dw}{2\pi
i}\frac{V^{\prime}(w)}{z-w}\sqrt{\frac{(z-a)(z-b)}{(w-a)(w-b)}}$ (126)

where $\mathcal{C}$ is a counter-clockwise contour around the branch cut
between $a$ and $b$. The spectral edges $(a,b)$ can be found using the
properties of the asymptotic expansion of $R_{0}(z)$. We can do the contour
integral (126) for the potential $V(w)$ given in equation (116). Since the
potential (69) is an even function of $\theta$, the eigenvalue density is
symmetric about $\theta=0$. As a result, in a complex $z$-plane the branch
cuts of $R(z)$ are symmetrically located on the unit circle about the real
axis. Hence we can take

$a=b^{-1}=e^{i\theta}.$ (127)

The final answer is given by,

$\displaystyle R_{0}(z)=\frac{1}{2z}-\frac{p}{4\pi\lambda
z}\log\left[\frac{z^{2}\left(\cos\theta-z+\sqrt{z^{2}-2\ z\
\cos\theta+1}\right)}{1-z\ \cos\theta+\sqrt{z^{2}-2\ z\
\cos\theta+1}}\right].$ (128)

It is easy to check that $R_{0}(z)$ has a branch cut singularity between
$\theta=\pm 2\sec^{-1}e^{\pi\lambda/p}$ on the unit circle in the complex $z$
plane. This implies the spectral edges are given by

$\displaystyle a=b^{-1}=\exp\big({2i\sec^{-1}e^{\pi\lambda/p}}\big{missing}).$
(129)

Using an expression analogous to (123), we can calculate the discontinuity of
$R_{0}(z)$ and find the leading eigenvalue density. The answer is given in
(72). The coefficient of $1/z^{2}$ in the asymptotic of $R_{0}(z)$ can be
computed and that matches with leading contribution of the expectation value
of $\mathscr{T}\mathrm{r}~{}U$ given in (83) for $p=1$.

Our next goal is to compute $R_{1}(z)$. The coefficient of $1/z^{2}$ in the
asymptotic expansion of $R_{1}(z)$ gives $1/N^{2}$ correction of
$\langle\mathscr{T}\mathrm{r}U\rangle$ viz.
$\langle\mathscr{T}\mathrm{r}U\rangle_{1}$. We follow the prescription given
in Ambjorn:1992gw  to write $R_{1}(z)$ in terms of a series of functions. We
shall not elaborate the method in this paper. $R_{1}(z)$ is given by

$\displaystyle R_{1}(z)$
$\displaystyle=\frac{1}{8(a-b)}\left(\psi^{(1)}(z)-\chi^{(1)}(z)\right)+\frac{1}{16}\left(\psi^{(2)}(z)+\chi^{(2)}(z)\right)$
(130)

where

$\displaystyle\chi^{(1)}(z)$ $\displaystyle=\frac{1}{M_{1}}\Phi_{a}^{(1)}(z),$
(131) $\displaystyle\psi^{(1)}(z)$
$\displaystyle=\frac{1}{J_{1}}\Phi_{b}^{(1)}(z),$ $\displaystyle\chi^{(2)}(z)$
$\displaystyle=\frac{1}{M_{1}}\left(\Phi_{a}^{(2)}(z)-\chi^{(1)}(z)M_{2}\right),$
$\displaystyle\psi^{(2)}(z)$
$\displaystyle=\frac{1}{J_{1}}\left(\Phi^{(2)}_{b}(z)-\psi^{(1)}(z)J_{2}\right).$

The functions $\Phi^{(n)}_{a}(z)$ and $\Phi^{(n)}_{b}(z)$ are given by

$\displaystyle\Phi^{(n)}_{a}(z)$
$\displaystyle=\frac{1}{(z-a)^{n}\sqrt{(z-a)(z-b)}},$ (132)
$\displaystyle\Phi^{(n)}_{b}(z)$
$\displaystyle=\frac{1}{(z-b)^{n}\sqrt{(z-a)(z-b)}}.$

$M_{n}$ and $J_{n}$ are given by the following contour integrals

$\displaystyle M_{n}$ $\displaystyle=\oint_{\mathcal{C}}\frac{d\omega}{2\pi
i}\frac{V^{\prime}(\omega)}{(\omega-a)^{n}\sqrt{(\omega-a)(\omega-b)}},$ (133)
$\displaystyle J_{n}$ $\displaystyle=\oint_{\mathcal{C}}\frac{d\omega}{2\pi
i}\frac{V^{\prime}(\omega)}{(\omega-b)^{n}\sqrt{(\omega-a)(\omega-b)}}.$

Evaluating the contour integrals (133) we find

$\displaystyle M_{1}$
$\displaystyle=\frac{4A}{a^{3/2}\left(\sqrt{a}+\sqrt{b}\right)},\qquad\quad\ \
J_{1}=\frac{4A}{b^{3/2}\left(\sqrt{a}+\sqrt{b}\right)}$ (134) $\displaystyle
M_{2}$
$\displaystyle=-\frac{4A\left(5\sqrt{a}+4\sqrt{b}\right)}{3a^{5/2}\left(\sqrt{a}+\sqrt{b}\right)^{2}},\qquad
J_{2}=-\frac{4A\left(4\sqrt{a}+5\sqrt{b}\right)}{3b^{5/2}\left(\sqrt{a}+\sqrt{b}\right)^{2}},$

where $A=-\frac{p}{4\pi\lambda}$. Plugging all these expressions in (130) we
finally arrive at

$\displaystyle R_{1}(z)=-\frac{\pi\lambda(1+z)(1-8z\cos\theta-6z+z^{2})\
\sin^{2}\frac{\theta}{2}}{12p\left(1-2z\cos\theta+z^{2}\right)^{5/2}}.$ (135)

We expand $R_{1}(z)$ asymptotically to obtain

$R_{1}(z)=-\frac{\pi\lambda\ \sin^{2}\frac{\theta}{2}}{12\ p\
z^{2}}+\frac{\pi\lambda\ \sin^{2}\frac{\theta}{2}(3\cos\theta+5)}{12\ p\
z^{3}}+\mathcal{O}\left(\frac{1}{z^{4}}\right).$ (136)

Thus from the asymptotic expansion we collect the coefficient of $1/z^{2}$ and
get

$\langle\mathscr{T}\mathrm{r}U\rangle_{1}=-\frac{\pi\lambda\
\sin^{2}\frac{\theta}{2}}{12\
p}=-\frac{\pi\lambda\left(1-e^{-\frac{2\pi\lambda}{p}}\right)}{12p}.$ (137)

Following the prescription (88) we find

$\langle\mathscr{T}\mathrm{r}U\rangle_{1}=\frac{i\pi\lambda\left(1-e^{2i\pi\lambda}\right)}{12}.$
(138)

The result (138) matches exactly with the subleading contribution of the
vertical framing $U(N)$ invariant of an unknot
$\mathcal{W}_{\Box}^{(2,1)}(S^{3},\lambda)$ carrying fundamental
representation on its component.

####  6.1.2 Two-loop correlator

We next compute the _leading_ connected piece of two point function
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle$. We define two-
loop correlator as

$R(z,w)=\frac{1}{N^{2}}\left\langle\Tr\frac{1}{z-U}\Tr\frac{1}{w-U}\right\rangle.$
(139)

In the large $N$ limit two-loop correlator is equal to the product of two one-
loop correlators $R(z)$ and $R(w)$. There is a connected subleading piece,
denoted by $R^{(c)}(z,w)$ and suppressed by a factor of $N^{2}$

$R(z,w)=R(z)R(w)+\frac{1}{N^{2}}R^{(c)}(z,w).$ (140)

Our goal is to find the connected piece of the two-point correlator. For the
partition function (113), the eigenvalue density can be written as

$\rho(z)=\left\langle\frac{1}{N}\sum_{i=1}^{N}\delta(z-z_{i})\right\rangle=-\frac{1}{N^{2}}\frac{\delta}{\delta
V(z)}\log\mathcal{Z}.$ (141)

Here we have used the fact that $\frac{\delta V(x)}{\delta V(y)}=\delta(x-y)$.
Repeating the similar analysis we can write

$\frac{\delta\rho(z)}{\delta
V(w)}=-N^{2}\left(\rho(z,w)-\rho(z)\rho(w)\right)$ (142)

where

$\rho(z,w)=\frac{1}{N^{2}}\left\langle\sum_{i,j=1}^{N}\delta(z-z_{i})\delta(w-z_{i})\right\rangle.$
(143)

In the large $N$ limit, $\rho(z,w)$ is also decomposed into a product of two
eigenvalue densities $\rho(z)$ and $\rho(w)$. The connected subleading
contribution $\rho_{c}(z,w)$ is suppressed by $1/N^{2}$ factor:

$\rho(z,w)=\rho(z)\rho(w)+\frac{1}{N^{2}}\rho_{c}(z,w).$ (144)

Hence we find

$\frac{\delta\rho(z)}{\delta V(w)}=-\rho_{c}(z,w).$ (145)

We know that using the definition of eigenvalue density, the one-loop
correlator $R(z)$ can be expressed as (120). After a little algebra we find
that

$\int\frac{dv}{w-v}\frac{\delta R(z)}{\delta V(v)}=-\int du\int
dv\frac{\rho_{c}(u,v)}{(z-u)(w-v)}.$ (146)

Using the definition (143) we can express two-point correlator as

$R(z,w)=\int du\int dv\frac{\rho(u,v)}{(z-u)(w-v)}.$ (147)

Likewise for the connected components of $R(z,w)$ and $\rho(u,v)$ we can write

$R^{(c)}(z,w)=\int du\int dv\frac{\rho_{c}(u,v)}{(z-u)(w-v)}.$ (148)

Thus we have

$\int\frac{dv}{w-v}\frac{\delta R(z)}{\delta V(v)}=-R^{(c)}(z,w).$ (149)

Using the analytic properties (122) of $R(z)$ about the cut we find that

$R^{(c)}(z+i\epsilon,w)+R^{(c)}(z-i\epsilon,w)=-\frac{1}{(z-w)^{2}}.$ (150)

Since $R(z,w)$ is symmetric in $z$ and $w$ similar property holds for the $w$
variable also. The connected piece of the resolvent, $R^{(c)}(z,w)$ being
discontinuous about the cut, $R^{(c)}(z,w)+\frac{1}{2(z-w)^{2}}$ must change
sign as we cross the cut for both the arguments. Therefore we take an ansatz
for $R^{(c)}(z,w)$ as follows

$R^{(c)}(z,w)=-\frac{1}{2(z-w)^{2}}\left(1-\frac{Q(z,w)}{\sqrt{\sigma(z)}\sqrt{\sigma(w)}}\right),\quad\text{where}\quad\sigma(z)=(z-a)(z-b).$
(151)

$Q(z,w)$ is symmetric meromorphic in its arguments. $Q(z,w)$ can be found for
a one-gap solution from the properties of $R^{(c)}(z,w)$. From the definition
of $R(z,w)$ we see that it is regular as $z\rightarrow w$. Expanding
$R^{(c)}(z,w)$ near $w=z$ we arrive at

$R^{(c)}(z,w)=-\frac{1}{2(z-w)^{2}}\left(1-\frac{Q(z,z)}{\sigma(z)}-\frac{(w-z)}{\sigma(z)}\left(\partial_{w}Q(z,w)-\frac{\sigma^{\prime}(w)Q(z,w)}{2\sigma(z)}\right)\bigg{|}_{w=z}\right).$
(152)

Therefore to make $R^{(c)}(z,w)$ finite at $z=w$ we have

$\displaystyle\lim_{w\rightarrow z}Q(z,w)$
$\displaystyle=\sigma(z)~{}~{}\text{and}~{}~{}\lim_{w\rightarrow
z}\partial_{w}Q(z,w)=\frac{\sigma^{\prime}(z)}{2}.$ (153)

$Q(z,w)$ must be regular at spectral edges hence $Q(z,w)$ must be a
polynomial. Since $R(z,z)$ goes as $1/z^{2}$ asymptotically, for one-cut
solution $Q(z,z)$ must be a polynomial of order 2. The above conditions can
uniquely determine $Q$ for one-cut solution and it is given by

$Q(z,w)=zw-\frac{1}{2}(a+b)(z+w)+ab.$ (154)

The leading contribution of
$\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}$
can be obtained from $\frac{1}{z^{2}w^{2}}$ coefficient of $R^{(c)}(z,w)$ in
its asymptotic expansion. After a careful computation we find that

$\bm{\ell}.\bm{c}.\left[\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\right]=\frac{(a-b)^{2}}{16}=e^{-\frac{2\pi\lambda}{p}}\left(e^{-\frac{2\pi\lambda}{p}}-1\right),$
(155)

where we have used the expressions of the spectral edges (129). With the
prescription (88) of analytic continuation $p\rightarrow ip$, followed by
setting $p=1$ we arrive at

$\bm{\ell}.\bm{c}.\left[\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\right]=e^{2i\pi\lambda}(e^{2i\pi\lambda}-1).$
(156)

The result (156) matches exactly with the following result from the HOMFLY-PT
polynomials:

$\displaystyle\mathcal{W}_{\Box,\Box}^{[(2,2),f=\\{1,1\\}]}(S^{3},\lambda)-\left(\mathcal{W}_{\Box}^{(2,1)}(S^{3},\lambda)\right)^{2}.$
(157)

This suggests that
$\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}$
evaluated in (155) is exact in $N$. Any further large $N$ corrections of
$\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}$
from the matrix model approach will vanish. Thus we have

$\displaystyle\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}$
$\displaystyle=e^{-\frac{2\pi\lambda}{p}}\left(e^{-\frac{2\pi\lambda}{p}}-1\right).$
(158)

###  6.2 Knot/link invariants and connected correlators

So far we have confined ourselves to Hopf link invariant with fundamental
representation placed on its components. Now we look at the connected piece of
two point correlator carrying some low dimensional representations apart from
the fundamental. Using the fact that Schur polynomials for any arbitrary
representation can be written as sum over different powers of $\Tr U^{m}$
(75), we can express the large $N$ results of any (torus) knot or link
invariant modulo its leading contribution in terms of connected correlation
functions of $\Tr U^{m}$. For example, consider the unknot invariant in
representation $\ydiagram[]{2}$, i.e.
$\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},\lambda)$.
Using (86) the leading contribution can be read off to be

$\displaystyle\bm{\ell}.\bm{c}.\left[\mathscr{V}^{(2,1)}_{\tiny{\ydiagram[]{2}}}(S^{3}/\mathbb{Z}_{p},\lambda)\right]$
$\displaystyle=\frac{1}{2}\left(\bm{\ell}.\bm{c}.\left[\mathscr{V}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},\lambda)\right]\right)^{2}.$
(159)

We know that (89, 75)

$\displaystyle\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)$
$\displaystyle=\langle\mathbf{s}_{\tiny{\ydiagram[]{2}}}\rangle,~{}~{}~{}\text{where}~{}~{}~{}\mathbf{s}_{\tiny{\ydiagram[]{2}}}=\frac{N}{2}\left(\mathscr{T}\mathrm{r}U^{2}\right)+\frac{N^{2}}{2}\left(\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right).$
(160)

Hence we have

$\displaystyle\mathscr{V}^{(2,1)}_{\tiny{\ydiagram[]{2}}}(S^{3}/\mathbb{Z}_{p},k)-\frac{1}{2}\left(\mathscr{V}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)\right)^{2}$
$\displaystyle=\frac{N}{2}\left\langle\mathscr{T}\mathrm{r}U^{2}\right\rangle+\frac{1}{2}\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}.$
(161)

This expression can also seen to be consistent using cabling prescription
ramadevi2000computation  and Frobenius relation in group theory
labastida2001knots ; marino2001framed ; ramadevi2012reformulated . In a
similar fashion
$\mathscr{V}_{\tiny{\ydiagram[]{1,1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)$ admits
a relation

$\displaystyle\mathscr{V}_{\tiny{\ydiagram[]{1,1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)-\frac{1}{2}\left(\mathscr{V}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)\right)^{2}=-\frac{N}{2}\langle\mathscr{T}\mathrm{r}U^{2}\rangle+\frac{1}{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}.$
(162)

Thus we see that when we change the representations, the unknot invariants
(modulo leading contribution) have different expansions in terms of
correlation functions of $\mathscr{T}\mathrm{r}U^{m}$ operators.

Similar analysis can be done for a two component Hopf link whose invariant
$\mathscr{V}^{(2,2)}_{\tiny{\ydiagram[]{2},\ydiagram[]{1}}}(S^{3}/\mathbb{Z}_{p},k)=\langle\mathbf{s}_{\tiny{\ydiagram[]{2}}}\mathbf{s}_{\tiny{\ydiagram[]{1}}}\rangle$
satisfies:

$\displaystyle\mathscr{V}^{(2,2)}_{\tiny{\ydiagram[]{2},\ydiagram[]{1}}}(S^{3}/\mathbb{Z}_{p},k)-\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)\mathscr{V}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)$
$\displaystyle=N\langle\mathscr{T}\mathrm{r}U\rangle\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
$\displaystyle+\frac{1}{2}\langle\mathscr{T}$
$\displaystyle\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}+\frac{1}{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U^{2}\rangle_{c}.$
(163)

In fact any $n$ point correlator can be expressed in terms of the connected
correlators as

$\displaystyle\langle\underbrace{\mathscr{T}\mathrm{r}U\cdots\mathscr{T}\mathrm{r}U}_{n\
\text{times}}$
$\displaystyle\rangle=\langle\mathscr{T}\mathrm{r}U\rangle^{n}+\frac{{}^{n}C_{2}}{N^{2}}\langle\mathscr{T}\mathrm{r}U\rangle^{n-2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
(164)
$\displaystyle~{}~{}~{}~{}+\frac{{}^{n}C_{3}}{N^{3}}\langle\mathscr{T}\mathrm{r}U\rangle^{n-3}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
$\displaystyle~{}+\frac{{}^{n}C_{4}}{N^{4}}\langle\mathscr{T}\mathrm{r}U\rangle^{n-4}\left({}^{4}C_{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}^{2}+\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\right)$
$\displaystyle+$
$\displaystyle\frac{{}^{n}C_{5}}{N^{5}}\langle\mathscr{T}\mathrm{r}U\rangle^{n-5}\left({}^{5}C_{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}+\langle\left(\mathscr{T}\mathrm{r}U\right)^{5}\rangle_{c}\right)$
$\displaystyle+\frac{{}^{n}C_{6}}{N^{6}}$
$\displaystyle\langle\mathscr{T}\mathrm{r}U\rangle^{n-6}\left({}^{6}C_{2}\langle\left(\mathscr{T}\mathrm{r}U\right)^{2}\rangle_{c}\langle\left(\mathscr{T}\mathrm{r}U\right)^{4}\rangle_{c}+^{6}C_{3}\langle\left(\mathscr{T}\mathrm{r}U\right)^{3}\rangle^{2}_{c}+\langle\left(\mathscr{T}\mathrm{r}U\right)^{6}\rangle_{c}\right)$
$\displaystyle+\cdots{}{}{}$ .

In order to validate the result with the colored HOMFLY-PT polynomial we
compute the leading contribution of (6.2) in the matrix model side. It is
given by (83,155)

$\displaystyle\bm{\ell}.\bm{c}.$
$\displaystyle\left[\mathscr{V}^{(2,2)}_{\tiny{\ydiagram[]{2},\ydiagram[]{1}}}(S^{3}/\mathbb{Z}_{p},\lambda)-\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},\lambda)\mathscr{V}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},\lambda)\right]=N\langle\mathscr{T}\mathrm{r}U\rangle_{0}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=Np\left(\frac{1-e^{-\frac{2\pi\lambda}{p}}}{2\pi\lambda}\right)e^{-\frac{2\pi\lambda}{p}}\left(e^{-\frac{2\pi\lambda}{p}}-1\right).$
(165)

Performing the analytic continuation $p\rightarrow ip$ followed by setting
$p=1$ (88) the above expression becomes identical to the one obtained from the
corresponding colored HOMFLY-PT polynomials:

$\displaystyle\bm{\ell}.\bm{c}.$
$\displaystyle\left[\mathcal{W}^{[(2,2);f=\\{1,1\\}]}_{\tiny{\ydiagram[]{2},\ydiagram[]{1}}}(S^{3},\lambda)-\mathcal{W}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3},\lambda)\mathcal{W}_{\tiny{\ydiagram[]{1}}}^{(2,1)}(S^{3},\lambda)\right]=\frac{2iN}{\pi\lambda}e^{4\pi
i\lambda}\sin^{2}\pi\lambda.$ (166)

We provide one more example before concluding this section.

$\displaystyle\mathscr{V}^{(2,2)}_{\tiny{\ydiagram[]{2},\ydiagram[]{2}}}(S^{3}/\mathbb{Z}_{p},k)-\left(\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},k)\right)^{2}=$
$\displaystyle
N^{2}\langle\mathscr{T}\mathrm{r}U\rangle^{2}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
(167) $\displaystyle+N\big{(}\langle\mathscr{T}\mathrm{r}U$
$\displaystyle\rangle\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}+\langle\mathscr{T}\mathrm{r}U\rangle\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U^{2}\rangle_{c}\big{)}$
$\displaystyle+\frac{5}{4}\langle\mathscr{T}$
$\displaystyle\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}^{2}+\frac{1}{4}\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$
$\displaystyle+\frac{1}{4}$
$\displaystyle\langle(\mathscr{T}\mathrm{r}U^{2})^{2}\rangle_{c}+\frac{1}{2}\langle(\mathscr{T}\mathrm{r}U)^{2}\mathscr{T}\mathrm{r}U^{2}\rangle_{c}.$

Again we can compute the leading contribution of (167) in the double scaling
limit using the matrix model results (83,155):

$\displaystyle\bm{\ell}.\bm{c}.$
$\displaystyle\left[\mathscr{V}^{(2,2)}_{\tiny{\ydiagram[]{2},\ydiagram[]{2}}}(S^{3}/\mathbb{Z}_{p},\lambda)-\left(\mathscr{V}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3}/\mathbb{Z}_{p},\lambda)\right)^{2}\right]=N^{2}\left\langle\mathscr{T}\mathrm{r}U\right\rangle_{0}^{2}\left\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\right\rangle_{c}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=N^{2}p^{2}\left(\frac{1-e^{-\frac{2\pi\lambda}{p}}}{2\pi\lambda}\right)^{2}e^{-\frac{2\pi\lambda}{p}}\left(e^{-\frac{2\pi\lambda}{p}}-1\right).$
(168)

Now we analytically continue $p\rightarrow ip$ and set $p=1$ (88) to get

$\bm{\ell}.\bm{c}.\left[\mathcal{W}^{[(2,2);f=\\{1,1\\}]}_{\tiny{\ydiagram[]{2},\ydiagram[]{2}}}(S^{3},\lambda)-\left(\mathcal{W}_{\tiny{\ydiagram[]{2}}}^{(2,1)}(S^{3},\lambda)\right)^{2}\right]=\frac{2iN^{2}}{\pi^{2}\lambda^{2}}e^{5\pi
i\lambda}\sin^{3}\pi\lambda.$ (169)

We see from equations (109, 6.2, 167) that as we change the representations
associated with two of the components of Hopf link, the expansion in terms of
correlation functions also get modified. Therefore any unknot or Hopf link
invariant (modulo the leading contribution) with arbitrary representations
placed on the components can be uniquely expressed in terms of different
connected correlation functions. For example, from equations (161) and (162)
we see that unknots in representations $\ydiagram[]{2}$ and $\ydiagram[]{1,1}$
are expressed in terms of $\langle\mathscr{T}\mathrm{r}U\rangle$ and
$\langle\mathscr{T}\mathrm{r}U\mathscr{T}\mathrm{r}U\rangle_{c}$. However they
differ by the sign of $\langle\mathscr{T}\mathrm{r}U\rangle$ term. Using the
character expansion for Schur polynomial (75) and equation (164) one can
similarly express any $(L,L)$ torus link invariant uniquely in terms of
connected correlation functions.

It is also to be noted that the evaluation of the corresponding leading
contributions as in equations (165, 167) using the matrix model results (83,
155) was possible because we have placed low dimensional representations on
the two component knots of the Hopf link. If we put arbitrary large
representations it will be difficult to evaluate the explicit expressions of
leading contributions with the matrix model techniques. This is because the
results (83, 155) involve the density distribution which extremises the
partition function of the manifold and any link carrying large representation
will back react on that solution.

Moreover, for any arbitrary $L$ component torus link the computation of the
connected correlator seems technically challenging from the matrix model side,
even with low dimensional representations placed on the component knots.
Because such invariants will involve higher point connected correlators, which
are difficult to handle. Nonetheless, for low representations the colored
HOMFLY-PT polynomial (31) allows us to obtain the subleading contribution for
any $L>2$. For example the polynomials tabulated in Table LABEL:Tableof3,3lc
will help us to get the subleading contribution to the $(3,3)$ torus link
invariant.

##  7 Conclusion

In this paper, we have discussed the torus link invariants in $U(N)$ Chern-
Simons theory embedded inside three manifold $S^{3}$ as well as
$S^{3}/\mathbb{Z}_{p}$. In the double scaling limit (1), it has been shown
that any $(L\alpha,L\beta)$ torus link invariant in $S^{3}$ (31) can be
expressed in terms of $(L,L)$ torus link invariant, here $\alpha$ and $\beta$
are coprime to each other. Our result is depicted in equation (54). Using the
explicit expression of the $(L\alpha,L\beta)$ invariant (31) we can evaluate
the polynomial for some low dimensional representations placed on the $L$
component knots of the link (e.g. see Table LABEL:Tableof3,3lc). Consequently,
in the large $N$ limit, we can deduce their leading and subleading
contributions. However, for links carrying higher dimensional representations
this procedure seems practically challenging. Matrix model techniques seem to
provide some handle of such difficulties at certain instances. For example,
when the representations placed on the component knots of any $(L,L)$ torus
link (89) is large (i.e. comparable to rank $N$ of the gauge group in the
limit $N\rightarrow\infty$) but completely symmetric we can determine the
leading contribution (103) to the invariant by using the eigenvalue density
(72) that dominates partition function (56) of the manifold
$S^{3}/\mathbb{Z}_{p}$. Moreover, for low dimensional representations, using
the aforementioned density distribution, we have written down an analytic
expression of the leading contribution of any $(L,L)$ torus link invariant
(87). The leading contribution of the torus link invariant being proportional
to the leading contribution of the unknot invariant, it does not capture the
intertwining information of the component knots constituting the link. To
capture such an information the study of connected piece of the correlators in
Chern-Simons theory become necessary. From the matrix model approach, we
analyse in detail the connected piece of a two point correlator (108), which
is basically the subleading contribution to the Hopf link invariant when both
the component knots carry fundamental representations (109). Our detailed
analysis lead to the result in equation (155) and it matches exactly with the
one obtained using the HOMFLY-PT polynomials (156). We also obtain the
subleading contribution to the unknot invariant carrying fundamental
representation (137), using the techniques of matrix model. It is interesting
to see that if we place low dimensional representations other than fundamental
then the Hopf link or unknot invariant modulo the corresponding leading
contribution take different structures expressed in terms of correlators of
$\mathscr{T}\mathrm{r}U^{m}$ operators as shown in equations (161, 162, 6.2,
167). As a validation, we specifically evaluate the leading contribution of
(6.2) and (167) using our matrix model computations (83, 155) and find the
result to agree with the respective colored-HOMFLY-PT polynomials.

Note that, in the double scaling limit Chern-Simons theory admits a third
order phase transition similar to Douglas-Kazakov phase transition
Chakraborty:2021oyq . This means the integrable representation (72) that
dominates the partition function (56) changes at some critical value of
$\lambda$. As a result the correlation functions when evaluated on the leading
solution must change its behaviour at the critical value. However, the origin
of such behaviour is not clear from the perspective of HOMFLY-PT polynomials.

###### Acknowledgements.

The work of AM is supported in part by the Prime Minister’s Research
Fellowship provided by the Ministry of Education, Government of India. The
work of KC is supported in part by the South African Research Chairs
Initiative (SARChI) of the Department of Science and Innovation and the
National Research Foundation. PR would like to acknowledge
SPARC/2019-2020/P2116/ project funding. PR would also like to acknowledge the
ICTP’s Associate programme where some work on this project had happened during
her visit as senior associate.

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