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

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.

4 Torus link invariants in the double scaling limit of U (N ) Chern-Simons theory 9 4.1 (2, 2m) torus link invariant in the double scaling limit 10 4.2 (Lα, Lβ) torus link invariant in the double scaling limit 11 5 Torus link invariants and matrix models at large N 13 5.1 Leading contribution to (L, L) torus link invariant for small representations 15 5.2 Leading contribution to (L, L) torus link invariant for large symmetric representations 20 6 Subleading corrections to Hopf link invariant 22 6.1 Calculation of ⟨TrU ⟩ 1 and ⟨TrU TrU ⟩ c 24 6.1.1One-loop correlator 24 6.1.2Two-loop correlator 27 6.2 Knot/link invariants and connected correlators 29 1 Introduction The famous discovery of Jones polynomial [1,2] by Vaughan Jones in 1984 led to a rejuvenation of the mathematical theory of knots and links.A knot K is an embedding of a circle S 1 inside a three-manifold M whereas a link L is a collection of two or more knots linked in a nontrivial way.Two knots K 1 and 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 [3], Jones polynomial, HOMFLY-PT polynomial [4,5] 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 [6].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 R ≡ of SU (2).Likewise, with the fundamental representations of G = SU (N ) and SO(N ) we obtain HOMFLY-PT polynomial and Kauffman polynomial [7] 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 (Lα, Lβ) where Lα and Lβ denote the number of times the link wraps around the meridian and the longitude of the torus respectively.Here α and β are coprime to each other.These torus links can be obtained as a closure of Lα strand braid with (Lα − 1)Lβ 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 where k is the coupling constant of the Chern-Simons theory.In terms of the variable q = exp 2πi k+N the above limit can be rephrased as q → 1, N → ∞ such that q N = fixed. (1.2) 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 [8].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 (Lα, Lβ) 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. [8] 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 [8,9].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. [10,11] 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) [12].
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 [8].The next subsection 4.2 contains the generalisation of this result for any L component (Lα, Lβ) 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.

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 [13]: 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 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 K in an irreducible representation R, the Wilson loop operator is defined as where, are the holonomies of SU (N ) and U (1) gauge connection respectively.The symbol P denotes path ordering.Note that the representation R ∈ U (N ) involves representation R ∈ SU (N ) and charge n ∈ U (1).Given (R, n), we can define U (N ) representation R in the following way: 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 (R) and s is any integer.The total number of boxes of representation R, known as the charge of U (1) theory, is given by n = N i=1 n i .The relation between SU (N ) Chern-Simons theory to su(N ) k Wess-Zumino Witten conformal field theory implies that the l i 's of the SU (N ) representations must obey l i ≤ 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 L made up of L number of component knots K a each carrying representation R a of U (N ) gauge group, then the Wilson loop operator of L is The correlation function of this topological operator yields the link invariant of L corresponding to U (N ) Chern-Simons theory These link invariants can be computed using one of the two framings: • standard framing where the self-linking numbers of the component knots are zero, • 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 [L; f ] where the framing f = {f 1 , • • • , 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 [L; f ] is given as where lk ab denotes the linking number between the components K a and K b .For the polynomial invariant of U (N ) Chern-Simons theory to be a function only of two variables in the U (1) invariant (2.8) we need to make the following replacement for the charge n (a)  and the coupling constant k 1 [13,14] We will elaborate the salient features of computing the (Lα, Lβ) 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 (Lα, Lβ) where α and β are coprime to each other.These links can be obtained from the closure of a braid with Lα strands.The braid word for a right handed (Lα, Lβ) torus link, in terms of the braid generators b i , is The corresponding mirror image of this link is left handed and its braid word becomes b   Given a braid word (3.1), there is a systematic procedure of directly writing down the invariants of links that are obtained from the closure of such braids [15].Basically, we use the basis where the braiding generators are diagonal.As any two consecutive generators b i and b i±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. 1a), closure of a simple two-strand braid with braid word b 2m 1 , 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 [16] λ where ϵ R 1 , ϵ R 2 , ϵ Rt = ±1, q is defined in equation (2.9) and C Ra , the quadratic Casimir of representation R a is defined as i denotes the number of boxes in the ith row of representation R a and l (a) = i l (a) i .Using the above braiding eigenvalue (3.2), 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 [13]: where the q-dimension of a representation is defined as The U (N ) invariant of framed (2, 2m) torus link can be written using (2.7, 2.8, 3.4): Note that, in writing the above expression we have used the fact that the quadratic Casimir and q-dimension of a U (N ) representation R a is same as that of the quadratic Casimir and q-dimension of SU (N ) representation R a : We introduce κ Ra in terms of the number of boxes n (a) i in the ith row of the Young diagram of representation R a : In terms of these variables, we express the U (N ) invariant of the framed (2, 2m) torus link as Hereafter we refer κ Ra to be the quadratic Casimir of U (N ) representation R a .The invariants for arbitrary L-component torus links can be compactly written using these quadratic casimirs κ Ra and q-dimension of representations [17].We will briefly present the explicit form of these link invariants in the following subsection.

(Lα, Lβ) torus link invariant
For a general two-component right handed torus link (2α, 2β) carrying U (N ) representations R 1 and R 2 on the component knots, the polynomial invariant is [17] W where ⊢n (1)   |c ) Here κ R in terms of κ R (3.9) can be written as The notation 3 , ... is a partition, in other words a sequence of positive integers with where ℓ(Λ (a) ) is the highest value of i for which Λ (a) i is non zero: The conjugacy class associated to the partition Λ (a) is denoted as c Λ (a) and it consists of one 3 -cycle and so on.|c Λ (a) | denotes the total number of elements belonging to the conjugacy class c Λ (a) , having the same cycle structure.χ Ra (c Λ (a) ) is the corresponding character labelled by representation R a .The sum of two partitions is the sum of each of the elements of the respective partitions and (Λ) (α) implies the partition (αΛ 1 , αΛ 2 , αΛ 3 , ...).
Rewriting the torus link invariant (3.11) in terms of variable κ R we get: 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 Now, we can generalise the colored HOMFLY-PT polynomial for an arbitrary L component torus link (Lα, Lβ), with self-linking number f = {f 1 , ...f L } of the component knots, as where In Ref. [8], two of the authors have analysed the double scaling limit (1.1,1.2) of the twocomponent (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 (Lα, Lβ).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 (1.1) of U (N ) Chern-Simons theory involves the Chern-Simons coupling k → ∞.The range of the n i 's (2.5) for the integrable representations R a at large k can be chosen as For the representation R a we define a set of N variables {θ N }: where, h It is easy to check that in the double scaling limit the variables θ (a) i range from −π to π.The quadratic Casimir of representation R a (3.9) can be expressed in terms of these variables as In the large N limit, the set of discrete variables θ i becomes a continuous function All the discrete summations over i are then replaced by integration over x Moreover the distribution of θ i , in the large N limit is captured by a distribution function such that dθσ(θ) = 1.Hence, for any arbitrary functions of θ i we can write 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 W ) can be expressed using the {θ i } variables (4.2) as Here θ i , θ (2) i and θ i are the set of variables corresponding to representations R 1 , R 2 and R t respectively.In the double scaling limit with appropriate replacement of the discrete variables in terms of the continuous functions (4.5, 4.6, 4.7) the above invariant becomes where is dependent on framing f and is independent of framing.Observe that the function G (f ) (m, λ, σ 1 , σ 2 ) depends only on the given representations R 1 and R 2 and is independent of R t .However, F(m, λ) involves the θ (t) integration which is constrained, since it runs only over the irreducible representations R t appearing in the tensor product of R 1 and R 2 .Therefore, this function F(m, λ) is difficult to evaluate even in the large N limit.Note that, the space of functional integration over θ (t) (x) in F(m, λ) remains the same for any value of m.As a result, the function F(m, λ) has the following property (S 3 , k) becomes the invariant of framed Hopf link.Therefore, by exploiting the property (4.13), we can see that a framed (2, 2m) torus link invariant can be expressed in terms of a framed Hopf link invariant as follows: That is, we first find the distribution functions σ 1 , σ 2 corresponding to the representations R 1 , R 2 placed on the two component knots.After which we can write the Hopf link invariant as a function of σ 1 , σ 2 and λ.Then the link invariant for (2, 2m) torus link as a function of λ becomes proportional to the Hopf link invariant with λ replaced by λ/m but importantly, keeping σ 1 and σ 2 fixed while performing such a replacement.
In the following subsection we generalise (4.14) for any L component torus link (Lα, Lβ) embedded inside S We now take the double scaling limit with the appropriate replacement of the discrete variables in terms of the continuous functions (4.5, 4.6, 4.7): where and When α = β = 1, we get the double scaling limit of Hopf link invariant.We exploit the symmetry of F β α , λ under to write down the invariant of any 2 component torus link in terms of Hopf link invariant as Similar exercise for any arbitrary L component torus link (3.19) is straightforward and we find that (Lα, Lβ) torus link invariant can be expressed in terms of (L, L) torus link invariant as follows: Thus, given a (L, L) torus link invariant in variable λ with σ 1 , ..., σ L being the eigenvalue densities of the component knots, the (Lα, Lβ) torus link invariant (carrying the same eigenvalue densities) in variable λ becomes proportional to the (L, L) torus link invariant with λ replaced by λ β/α while keeping σ 1 , ..., σ L unaltered.We can explicitly work out the (L, L) link invariants (3.19) 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 (Lα, Lβ) link invariant in the double scaling limit (4.22).Extracting such leading and subleading contributions appear to be practically difficult for representations with large number of boxes.In Ref. [8], 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 → ∞), 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 /Z p , which can equivalently be written in terms of the modular transformation matrices (S, T ) of u(N ) k Wess-Zumino conformal field theory.The explicit form of Hopf link invariant on S 3 /Z p with representations R 1 and R 2 on the two components is given by (5.1) The summation runs over all the irreducible integrable representations R of u(N ) k .Considering R 1 and R 2 to be trivial representations we obtain the partition function of Chern-Simons theory on (5.2) In the double scaling limit (1.1) the partition function is dominated by a single representation.Contributions due to other representations are suppressed by O(1/N 2 ).This is the standard saddle point approximation.The Hopf link invariant defined in (5.1) is unnormalised.We can define the normalised invariant as R 1 ,R 2 S 3 /Z p , k with the representations R 1 and R 2 being trivial becomes identically equal to one.Note that, with p = 1, In order to proceed further we explicitly write down the modular transformation matrix where for any integrable representations R a and R b ) Here, as before, n (a/b) and n denote the total number of boxes and the number of boxes in the i th row corresponding to the Young diagram of R a/b respectively.
The other modular transformation matrix T is given by Q R and c denote the conformal weight of representation R and central charge respectively Next we want to calculate the ratio of modular transformation matrices (5.9) After some calculation one can show that the ratio S RRa S 0R can be expressed as where the Schur polynomial s Ra (U ) of the unitary group element U in representation R a is j det e iθ i (N −j) . (5.11) Note that the notation U used in equation (5.10) corresponds to representation R on its left hand side i.e. the diagonal elements of U correspond to h i (4.3) of R. The unnormalised Hopf link invariant, therefore can be expressed as1 where q is the (k + N )th root of unity (2.9).In equation (5.12) each R corresponds to a unitary matrix U in an ensemble of unitary matrices.The summation over R therefore corresponds to summation over unitary matrices.The factor S 2 0R when written in terms of θ i variables, reduces to Vandermonde measure for unitary matrices.Thus in the large N limit we can write where we have ignored few θ independent overall factors which doesn't affect our analysis.
Hence the partition function (5.2) can be written as a 0-dimensional unitary matrix model with weight factor e −(N +k) 2 S[θ] as where (5.15) Note that, in order to arrive at the above result we have replaced p in equation (5.2) by −ip.The normalised Hopf link invariant (5.3) is therefore given as the integral of two Schur polynomials s R 1 (U ) and s R 2 (U ) over the unitary ensemble with the aforementioned weight factor (5.16) For large R 1 and R 2 (i.e. when the number of boxes n (1) i and n (2) i 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 R 1 and R 2 are small, one can find the leading order contribution to V (2,2) R 1 ,R 2 (S 3 /Z p , k) by computing s R 1 (U ) and s R 2 (U ) on the classical solution that extremizes the partition function (5.2).

Leading contribution to (L, L) torus link invariant for small representations
For a (2,2) torus link invariant when the "size" of R 1 and 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 s R 1 (U ) and s R 2 (U ) as probe and evaluate them on the background solution (5.18).
Let us define a distribution function for the variables {θ i } in the large N limit as (5.17) It was shown in Ref. [9] that in the double scaling limit (1.1) the partition function (5.2) is dominated by the following distribution function2 , with − 2 sec −1 e πλ/p < θ < 2 sec −1 e πλ/p (5.18) which is a solution to the following saddle point equation obtained from the partition function (5.14) Therefore in the probe approximation the Hopf link invariant (5.16) can be computed by evaluating the Schur polynomials on the solution (5.18) where Ū corresponds to the distribution (5.18).The Schur polynomial admits a character expansion [18] s where Since to each vector ⃗ k (a) we can associate a conjugacy class c( ⃗ k (a) ), the sum over ⃗ k (a) in (5.21) represents a sum over all possible conjugacy classes subjected to the condition (5.22).Thus in the probe approximation, computation of the Hopf link invariant reduces to computation of the quantities Υ ⃗ k (a) (U ) on the solution (5.18).
In the large N limit the expectation value of Tr U r is dominated by the background solution i.e where ρ r = cosrθ ρ(θ)dθ, (5.25) with ρ(θ) as given in (5.18).Therefore, in the large N limit s Ra ( Ū ) is given by where r . (5.26) For small representations k r 's are O(1) numbers.Hence z ⃗ k (a) 's are also O(1) numbers.As a result the N dependence of s Ra ( Ū ) is given by N K (a) .Therefore, in the large N limit s Ra ( Ū ) is dominated by a conjugacy class for which K (a) is maximum subjected to (5.22).Using (5.22) we can replace k (a) 1 in K (a) and find (5.27) Hence, the maximum value of K (a) is given by k where ρ 1 can be evaluated using (5.25) and is given by χ Ra 1 n (a) is the character of the conjugacy class consisting of n (a) number of 1-cycles in representation R a .Thus the leading contribution (denoted by ℓ.c.) to the unknot invariant in fundamental representation is given by 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 (5.20) n (1) !n (2) !(5.31) 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 □ (S 3 /Z p , λ) n (1) ! . (5.32) We can generalise (5.31) for (L, L) torus link invariant in the probe approximation and it is given by Ra (S 3 /Z p , λ) . (5.33) 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 1.Note that in order to match the matrix model result (5.33) with the tabulated ones we need to follow the prescription as mentioned below: • first, replace p → ip in the expressions obtained by matrix model calculations to compensate the replacement p → −ip done earlier (see below equation (5.15)), • then set p = 1, since the colored HOMFLY-PT polynomials are obtained for framed links embedded inside S 3 .
Explicitly, it implies that (5.34)When these modifications are taken into account, the expression of ρ 1 (5.29) becomes identical to the leading order contribution of the vertical framing U (N ) invariant W (2,1) □ (S 3 , λ) 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 1) are in exact agreement with those (5.33)obtained using the eigenvalue density (5.18) 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.

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 /Z p , carrying integrable representations R 1 , ..., R L on the component knots is given by Using the character expansion for Schur polynomial (5.21) and (5.13), in the double scaling limit we can express the above invariant as (5.36) where (5.37) For a given ⃗ k (1) , )] is dominated by a classical configuration satisfying the saddle point equation This is a coupled equation because the moments of the distribution ρ r = dθρ(θ)e irθ 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 ρ(θ) in the large N limit.But if any of the R a is large such that n (a) is of O(N 2 ), then following (5.22)some of the k r can also be of O(N 2 ).In such cases we can not neglect the last term to solve for ρ(θ).However, in the case of all representations (R a ) large but completely symmetric, there can be at most k boxes in the first row and hence n (a) ≤ k.As a result all the k (a) r 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 for large symmetric representations we can still use the probe approximation and calculate the Schur polynomials on the background solution (5.18).Thus the (L, L) link invariant (5.35) becomes s Ra ( Ū ). (5.39) where Ū corresponds to the density distribution solving (5.19).Note that, unlike (5.28) s Ra ( Ū ) 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 (5.26) can be written as, subjected to the condition r rk a) .We use the Lagrange multiplier α (a) to extremise the above summation.Suppose Extremising H (a) with respect to k 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 λ.However, we can find it for small λ.
For small λ the eigenvalue distribution (5.18) is given by (5.46) We can compute ρ r by using the formula (5.25) with ρ(θ) as defined above and range of θ lying from −2 2πλ/p to 2 2πλ/p.The result is as follows (5.47) Using (5.43) we can find that the Lagrange multiplier is given by, where κ (a) = n (a) N .Hence H (a) can be evaluated to obtain, (5.49) Substituting H (a) in equation ( 5.45) and then using (5.39) 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 1) 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.

Subleading corrections to Hopf link invariant
In this section we compute the subleading contribution to the invariant of Hopf link.From (5.16) 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.R 1 = R 2 = □.The character expansion of the Schur polynomial (5.21) tells us that s □ (U ) = Tr U .Thus the Hopf link invariant with both the components in fundamental representations can be expressed as Likewise, an unknot invariant in fundamental representation is given by We introduce a rescaled trace operator such that ⟨TrU m ⟩, when evaluated on saddle point gives a O(1) number, i.e. the leading contribution to ⟨TrU m ⟩ becomes of O (1).The corrections to ⟨TrU m ⟩ are suppressed by higher powers of N .In terms of rescaled trace operators, the Hopf link and unknot invariants can be written as We define the connected piece of a two-point function in the following way [10], where we have put an overall factor of N 2 on the right hand side such that the leading contribution of ⟨TrU TrU ⟩ c is again of O(1).This connected piece can be expressed in terms of the Hopf link and unknot invariants as In the large N limit ⟨TrU ⟩ admits the expansion where ⟨TrU ⟩ 0 ≡ ρ 1 (5.29) is the expectation value of TrU evaluated on the saddle point (5.18) and ⟨TrU ⟩ 1 is the subleading contribution.Therefore from (6.5) we see that the subleading contribution to the two point correlator 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 (6.4) as (6.9) The subleading correction to Hopf link invariant has two parts.⟨TrU ⟩ 0 ⟨TrU ⟩ 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 ⟨TrU TrU ⟩ 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 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 (6.14) we can write R(z) = dw ρ(w) z − w .(6.17) Using the Sokhotski-Plemelj theorem we find that the resolvent satisfies and Moreover R(z) has the following asymptotic Thus R(z) is the generating function for ⟨Tr U m ⟩ for different m > 0. The resolvent R(z) admits a perturbative large N expansion The leading contribution of the resolvent, R 0 (z) can be obtained by solving the Dyson-Schwinger equation [19,20] in the N → ∞ limit.R 0 (z) also has a branch cut on the unit circle.The eigenvalue density ρ(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 [11].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 [21] R 0 where 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 (6.23) for the potential V (w) given in equation (6.13).Since the potential (5.15) is an even function of θ, the eigenvalue density is symmetric about θ = 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

.24)
The final answer is given by, It is easy to check that R 0 (z) has a branch cut singularity between θ = ±2 sec −1 e πλ/p on the unit circle in the complex z plane.This implies the spectral edges are given by a = b −1 = exp 2i sec −1 e πλ/p .(6.26) Using an expression analogous to (6.20), we can calculate the discontinuity of R 0 (z) and find the leading eigenvalue density.The answer is given in (5.18).The coefficient of 1/z2 in the asymptotic of R 0 (z) can be computed and that matches with leading contribution of the expectation value of Tr U given in (5.29) 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 ⟨TrU ⟩ viz.⟨TrU ⟩ 1 .We follow the given in [11] 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 b (z), (6.28) The functions Φ M n and J n are given by the following contour integrals Evaluating the contour integrals (6.30) we find .40) In the large N limit, ρ(z, w) is also decomposed into a product of two eigenvalue densities ρ(z) and ρ(w).The connected subleading contribution ρ c (z, w) is suppressed by 1/N 2 factor: ρ(z, w) = ρ(z)ρ(w) + 1 N 2 ρ c (z, w).(6.41) Hence we find δρ(z) δV (w) = −ρ c (z, w).(6.42) We know that using the definition of eigenvalue density, the one-loop correlator R(z) can be expressed as (6.17).After a little algebra we find that

.43)
Using the definition (6.40) we can express two-point correlator as

.44)
Likewise for the connected components of R(z, w) and ρ(u, v) we can write Using the analytic properties (6.19) of R(z) about the cut we find that (6.47) 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) + 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 , where σ(z) = (z − a)(z − b).(6.48) 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 → w.Expanding R (c) (z, w) near w = z we arrive at 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 − 1 2 (a + b)(z + w) + ab.(6.51) The leading contribution of ⟨TrU TrU ⟩ c can be obtained from 1 z 2 w 2 coefficient of R (c) (z, w) in its asymptotic expansion.After a careful computation we find that For example, consider the unknot invariant in representation , i.e.V (2,1) (S 3 /Z p , λ).

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 /Z p .In the double scaling limit (1.1), it has been shown that any (Lα, Lβ) torus link invariant in S 3 (3.19)can be expressed in terms of (L, L) torus link invariant, here α and β are coprime to each other.Our result is depicted in equation (4.22).Using the explicit expression of the (Lα, Lβ) invariant (3.19) we can evaluate the polynomial for some low dimensional representations placed on the L component knots of the link (e.g.see Table 1).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 (5.35) is large (i.e.comparable to rank N of the gauge group in the limit N → ∞) but completely symmetric we can determine the leading contribution (5.49) to the invariant by using the eigenvalue density (5.18) that dominates partition function (5.2) of the manifold S 3 /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 (5.33).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 (6.5), which is basically the subleading contribution to the Hopf link invariant when both the component knots carry fundamental representations (6.6).Our detailed analysis lead to the result in equation ( 6.52) and it matches exactly with the one obtained using the HOMFLY-PT polynomials (6.53).We also obtain the subleading contribution to the unknot invariant carrying fundamental representation (6.34), 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 TrU m operators as shown in equations (6.58, 6.59, 6.60, 6.64).As a validation, we specifically evaluate the leading contribution of (6.60) and (6.64) using our matrix model computations (5.29, 6.52) 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 [9].This means the integrable representation (5.18) that dominates the partition function (5.2) changes at some critical value of λ.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.

−1 1 b −1 2 ...b − 1 Lα− 1 Lβ
. As an illustration, we show in figure (1) a two component (L = 2) torus link (with α = 1, β = 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.For clarity, we now discuss the colored HOMFLY-PT polynomial for these (2, 2m) torus links before getting into arbitrary L component torus links.

4. 2 (
Lα, Lβ) torus link invariant in the double scaling limit Let us first focus on arbitrary 2 component torus links (2α, 2β).Using the expression of κ Ra in terms of the set of N variables θ (a) i (4.4) we can express the (2α, 2β) torus link invariant (3.18) as becomes a Hopf link (3.10) embedded inside three manifold S 3 with framing of component knots f = {1, 1}.

S
RRa S 0R appearing in equation (5.1).Instead of representing a Young diagram by the set of box numbers {n (a) i } we use the variables θ i defined in equation (4.2).Since in the double scaling limit θ i ranges between −π and π, we can arrange them as eigenvalues of an N × N unitary matrix U = diag(e iθ 1 , • • • , e iθ N ).

( 3 •
Tr U r ) k (a) r , z ⃗ k (a) = r k (a) r !r k (a) r .(5.23) n (a) being the total number of boxes in representation R a .c( ⃗ k (a) ) ≡ 1 k (a) 1 2 k (a) 2 3 k (a) • • is the conjugacy class of the symmetric group G n (a) of degree n (a) , consisting of k 2 and so on.In other words, k (a) r denotes the number of r-cycle of the conjugacy class c( ⃗ k (a) ).χ Ra (c( ⃗ k (a) )) is the corresponding character, in representation R a .

6 . 2
ℓ.c.[⟨TrU TrU ⟩ c ] = (a − b)where we have used the expressions of the spectral edges(6.26).With the prescription (5.34) of analytic continuation p → ip, followed by setting p = 1 we arrive at ℓ.c.[⟨TrU TrU ⟩ c ] = e 2iπλ (e 2iπλ − 1).(6.53)The result (6.53) matches exactly with the following result from the HOMFLY-PT polynomials: ⟨TrU TrU ⟩ c evaluated in (6.52) is exact in N .Any further large N corrections of ⟨TrU TrU ⟩ c from the matrix model approach will vanish.Thus we have ⟨TrU TrU ⟩ c = e 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 (5.21), 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 .