Knot-Quiver correspondence for double twist knots

We obtain a quiver representation for a family of knots called double twist knots $K(p,-m)$. Particularly, we exploit the reverse engineering of Melvin-Morton-Rozansky(MMR) formalism to deduce the pattern of the charge matrix for these quivers.


I. INTRODUCTION
Knot-quiver correspondence (KQC) conjectured by Kucharski-Reineke-Stosic-Sulkowski [1] provides a new encoding of HOMFLY-PT invariants of knots in terms of the representation theory of quivers.Such a correspondence was motivated by studying the supersymmetric quiver quantum mechanics description of BPS states in brane systems describing knots [2].
Quivers are denoted as directed graphs with finite number of vertices connected by oriented edges.For a * vks2024@nyu.eduquiver with n number of vertices, the directed graph is encoded in a n × n quiver matrix C. The diagonal elements C ii refer to the number of loops at the 'i'-th vertex, and the off-diagonal elements C ij give the number of oriented edges from the vertex 'i' to the vertex 'j'.Hence, the elements in the matrix C are non-negative integers.
According to the conjecture, at least one quiver graph Q K is associated with the knot K satisfying the exponential growth property of the symmetric r-colored HOMFLY-PT polynomials P K r (A, q) (normalized appropriately) 1 as elaborated in the Refs.[1,3,4].In the context of KQC, the quivers are symmetric quivers.That is., ji .Further, these quiver matrix elements can be negative integers and be made non-negative by change of framing.Particularly, for the knot K which obey exponential growth property, we can write the generating series for P K r (A, q) in two equivalent forms: to extract the quiver matrix C (K) as well as the motivic Donaldson-Thomas (DT) invariants Ω d,j .Here d ≡ (d 1 , d 2 , . . .d n ) ≥ 0 and x d = n i=1 x di i with x i = xA βi q αi−1 (−1) γi .Note that sets {d i } satisfy the condition r = d 1 + d 2 + . . .+ d n .The procedure to obtain such a motivic series for any knot is still an open question.
For a class of torus knots (2, 2p + 1), the twist knots K p (p ∈ Z) and other knots upto seven crossings the quiver presentation were obtained [1,5].Except for unknot and trefoil, knots have more than one quiver presentation with the same number of nodes, indicating that the correspondence of knots to quivers is not unique.In Ref. [6], equivalent quivers with the same number of nodes were shown as vertices on a permutohedra graph, giving a systematic enumeration of such equivalent quivers.There are also quivers with different number of nodes that describe the same physics, i.e. a pool of dualities in 3d N = 2 theory [7].
The physical and the geometrical interpretation of the conjectural KQC was addressed [8] within the framework of Ooguri-Vafa large N duality.From the physics perspective, the motivic generating series of Q K matches the vortex partition function of 3d On the geometrical side [8], the spectrum of holomorphic curves with boundary on the conormal Lagrangian L K of the knot in the resolved conifold encodes the quiver data.That is., the basic holomorphic disks correspond to the nodes of the quiver Q K and the linking of their boundaries to the quiver arrows.
With double fat diagram description for arborescent knots, the r-colored HOMFLY-PT invariant: for a universal arborescent knot family K involving seven parameters was proposed in [9,10].Such a family includes most of the arborescent knots upto 10 crossings.Also, the colored HOMFLY-PT polynomials (upto four symmetric colors for knots upto 10 crossings) have been updated in the website [11].Recent papers on the existence of quivers for all rational knots, tangles and arborescent knots [5,12] motivated us to deduce a universal quiver for our arborescent knot family.Even though the problem is concrete, finding explicit quivers for this universal arborescent family appears to be a hard problem.
As a first step, we wanted to investigate some arborescent knots whose Alexander polynomial has a structure similar to that of the twist knots K p .That is., ∆(X) = 1 − pX.In fact, there is a systematic reverse engineering approach of the Melvin-Morton-Rozansky (MMR) formalism to obtain the quiver representation for such twist knots [13].We observed ∆(X) = 1 − 4X for knot 8 3 is part of the family of double twist knots characterized by two variables, denoted as K(p, −m), illustrated in Figure 1.Note p, m ∈ Z + denote the number of full-twists and the Alexander polynomial is ∆(X) = 1 − pmX.Such a form motivated us to attempt quiver representation for double twist knots.
Even though the r-colored HOMFLY-PT for any double twist knot in the cyclotomic form are known [14,15], rewriting them in the form of motivic series is still a challenging problem.We tried to determine the quiver representation of P K(p,−m) r (A = q N , q) following the methodology in Ref. [13].However, we faced computational difficulty in deducing the A dependence in the quiver representation.Also, we know that the quiver matrix C (K)   do not depend on the variable A = q N .As our aim is to conjecture the quiver matrix form for the double twist knots K(p, −m), we focus on rewriting r-colored Jones polynomial (A = q N =2 ) : J r (K(p, −m), q) ≡ P K(p,−m) r (A = q 2 , q) , as a motivic series.Particularly, we obtain quiver matrix C K(p,−m) i,j associated with the K(p, −m) for m ≤ 3.
We conjecture that the quiver matrix C K(m,−m) i,j is sufficient to recursively generate the quiver matrix for all the double twist knots K(p ̸ = m, −m).
We follow the route of reverse engineering of MMR expansion [13] to derive the motivic series form for . We will now briefly review the reverse engineering formalism, which will set the notation and procedure we follow for K(p, −m) in the next section.

I.1. Reverse Engineering of Melvin-Morton-Rozansky (MMR) expansion
Melvin-Morton-Rozansky(MMR) expansion states that the symmetric r-colored HOMFLY-PT for knot K has the following semiclassical expansion: lim ℏ→0,r→∞ with the leading term being the Alexander polynomial ∆ K (x) and the variable x in terms of color r is x = q r = const.The symbol R K k (x, N ) represent polynomials in the variable x.The reverse approach is to obtain P K r (A, q) using the Alexander polynomial ∆ K (x) [13].This approach also has obstacles to lift the ℏ → 0 expansion to q-dependent P K r (A, q) but can be fixed for some situations by comparing with the data of symmetric rcolored HOMFLY-PT polynomials known for r = 1, 2, 3. We will briefly highlight the steps involved in the reverse engineering formalism of MMR expansion [13]: i.We rewrite the Alexander polynomial in new variable X = (1−x) 2 x .Thus, the Alexander polynomial takes the following form: where the coefficients a i are integers, and s is a positive integer.
ii.Now, we use the following inverse binomial theorem to write the first term of MMR expansion (2) as follows: iii.We make the following quantum deformation to get the quantum-deformed polynomial: Here, the variable t is known as the refined parameter.In this article, we will take t = −1 to obtain unrefined polynomial invariants for double twist knots.The term within parentheses represents the q-Pochhammer, while square brackets correspond to the q-binomials, which are defined as: k depends on the knot K and must be written in terms of q-Pochhammers, q-binomials, and (q, A)-dependent powers so that r k q −rk (Aq r ; q) k cK k can be transformed into the following form to deduce the corresponding quiver Q K : Here C K i,j is the quiver matrix and the variables α i , β i and γ i are integer parameters.The set Even though such a transformation is motivated by comparing Ooguri-Vafa partition function [16] with the motivic generating series [17][18][19], it is still a hard problem to obtain cK k for any knot.Note that the quadratic power of q depends on C K i,j and it is independent of A = q N .Hence, we will work with the colored Jones polynomials J r (K, q) of a knot K to extract its quiver matrix using the reverse engineering techniques of MMR formalism replacing A → q2 in eqn.(4) 2 .
The plan of the paper is as follows: In section II.1, we briefly discuss the colored Jones polynomials of double twist knot K(p, −m) obtained from the reverse engineering techniques of MMR expansion.In section III, we conjecture C K i,j for K = K(p, −m) and validate it for some double twist knots.We conclude in section IV summarising our results, enumerating some open questions and future directions.

II. DOUBLE TWIST KNOTS
We have listed some of the double twist knots K(p, −m) in Table 1.As these double twist knots belong to arborescent family, the symmetric r-colored HOMFLY-PT polynomials can be obtained for every r from Chern-Simons theory [20,21].In fact, colored HOMFLY-PT for arbitrary r in closed form is given in Ref. [14].Hence, our aim is not to reconstruct r-colored HOMFLY-PT for double twist knots.We will now present the reverse engineering of MMR formalism (2) to rewrite r-colored Jones as a motivic series to extract the matrix of the quiver Q K(p,−m) .For given positive integers p and m, the Alexander polynomial of a double twist knot of type K(p, −m) takes the form Here . Such a linear expression appeared in many knots [13], suggesting the inverse binomial expansion to take the following form: Further, using the quantum deformation procedure discussed in [13] and taking A → q 2 (N = 2) in eqn.( 6), the colored Jones polynomial can be written as is (4pm + 1) × (4pm + 1) matrix for quiver Q K(p,−m) .It is worth noting that ξ i = α i + 2β i and γ i are integer parameters that can be determined by comparing them with r = 1, 2, 3 [14,22].By this approach, we explicitly determined {ξ i }, {γ i } parameters (7) for K (2,−2) = 8 3 knot: (−1) i γidi (q 2 ; q 2 ) r q i,j Ci,j didj 17 i=1 (q 2 ; q 2 ) di q ξidi , where To give clarity to the readers, we present a step-by-step procedure for determining the quiver matrix for the knot 8 3 in the Appendix.The polynomial invariants matches with the closed form [14] for large value of r as well confirming that the above 8 3 quiver data is indeed correct.Such an exercise for K(2, −2) suggested that we could propose and conjecture C K(p,−m) for the double twist knot family.We discuss them in the following section.We observe that the quiver matrix has a block structure by performing a similar analysis of the previous section for other examples of the double twist knots K(p, −m).Our explicit computation suggests the following proposition.

Proposition:
The r-colored Jones polynomial for double twist knots K(p, −m), with p ≥ m, can be expressed in the quiver representation: where the linear term The block structure of the matrix C K(p,−m) for some examples lead to the following conjecture: Conjecture: The generic structure of the quiver matrix will take the form where X ⊤ stands for transposition of matrix X, the row matrices of size 1 × 2m and F 0 = 0. Let X k denote the following set of 2m × 2m matrices : where k = 1, 2 . . .p.All these matrices can be recursively obtained using where J is a matrix of size 2m×2m where all the elements are one.So, knowing the set X 1 is sufficient to determine the full quiver matrix C K(p,−m) .It appears that the set X 1 for the simplest twist knot K(p = 1, −m = −1) ≡ 4 1 will suffice to obtain X 1 for double twist knots K(p, −m) as K(p, −m) = K * (m, −p) (K * denotes mirror image of the knot K).However, our matrix conjecture assumes p ≥ m.Hence, our explicit computations of set X 1 for m = 2, 3 is not derivable from the C K(p,−1) .
In the following subsections, we will give some examples to validate our proposition and conjecture.Specifically, we work out the X 1 set matrices for double twist knots K(m, −m) for m = 1, 2, 3.This set is sufficient to obtain the explicit quiver presentations for all the double twist knots K(p, −m) where m = 1, 2, 3.
Our matrix form for p = 3 is consistent with our conjecture (9), and the basic set of matrices X 1 are: Fk = 0, 0, 0 0 , and F 0 = (0).We further worked out for p = 4, 5 as well and verified our conjecture (9) form obeyed. From these computations, we can deduce the general form of the linear term Ξ (p,−2) and phase factor Λ (p,−2) in the proposition (8) for arbitrary p as: where and the phase factor is Using the above data, we can write the colored Jones polynomial for any K(p, −2) in quiver presentation with the quiver matrix consistent with the conjecture (9).So far, we have obtained the set of matrices X 1 for m = 1, 2.
With the hope of deducing gthe pattern for the set X 1 for any m, we will investigate double twist knots with m = 3 in the following subsection.
We have verified that our conjecture ( 9) is true for p = 4, 5.The linear term and phase factor in the proposition (8) for p = 3, 4 are as follows: Probably,there is a closed-form expression for Ξ p,−3 and Λ p,−3 for any p.We are not able to infer the closed form from the above data.
Ideally, it would be beneficial to find the set of matrices X 1 for any m as well as the closed form for Ξ p,−m and Λ p,−m .The size of the quiver matrix (4mp+1)×(4mp+1) makes the computations difficult.

IV. CONCLUSION AND DISCUSSION
Double twist knots K(p, −m) depend on two full twist parameters p, m ∈ Z + belong to the arborescent family (see Fig. 1).Finding a quiver with matrix (9) associated to each of the double twist knots was attempted using reverse engineering of Melvin-Morton-Rozansky expansion.We observed the Alexander polynomial form to be ∆(X) = 1−pmX, almost similar to twist knots K(p, −1) studied in Ref. [1].Comparing the structure of twist knot quiver, we put forth a proposition (8) for colored Jones in a quiver presentation as well as conjectured (9) the structure of the quiver matrix C K(p,−m) for any double twist knot K(p, −m).We have explicitly worked out some double twist knots to validate our proposition and the conjecture for m = 1, 2, 3. Our detailed methodology shows the complexity of the equations to deduce a concise form for Ξ p,−3 and Λ p,−3 .
We did attempt rewriting the superpolynomial for double twist knots K(p, −m) [15] in the quiver representation form using the q-multinomial identities [1].Unfor-tunately, we face computational difficulty even in determining the powers of A dependence.That is., β 1 , β 2 , . . . in the motivic series (1).We are working on an alternative method involving homological diagrams to obtain r-colored HOMFLY-PT in quiver representation [23].We hope to report these results soon [23].
It will be worth investigating whether our conjectured form for the block structure in C K(p,−m) has any connections to (i) the tangle operations and (ii) the holomorphic discs on the knot conormal.Probably, this could be another way to tackle quiver representation form for r-colored HOMFLY-PT.
Three-manifold invariants F K (x = q r , q) for knot complements S 3 \K can be deduced from the coefficients of the MMR expansions for r-colored Jones polynomial [24].Subsequently, this invariant was refined in Ref. [25].Similar to the knot-quiver correspondence, a motivic series for the F K (x, q) invariant was conjectured in Ref. [26].Such a conjecture has been validated for the torus knots of type (2,2p+1) [26], the double twist knots of type K(p, m), K(p + 1 2 , −m) [27].The large color behaviour of colored Jones polynomial was the starting point to tackle F K (x, q) for positive braid knots in Ref. [27].Further, the refined version of these three-manifold invariant for the knot complements of positive braids is discussed in Ref. [28].We still face a stumbling block in achieving a quiver form for these double twist knots of type K(p, m), K(p + 1 2 , −m) and the motivic series F K(p,−m) (x, q).We probably need a radical approach to obtain such a motivic series form.We will report on these aspects and their refined version in future.
There is a pretzel family of knots whose r-colored Jones and HOMFLY-PT are known.It will be interesting exercise if we can explicitly write a quiver presentation and deduce the matrix for the pretzel.From our double twist knot results, it may be straightforward to attempt quiver presentation for knots whose Alexander polynomial takes the form ∆(X) = 1 ± (m 1 m 2 . . .m p )X .We hope to address these problems in future.

ACKNOWLEDGMENTS
The work of VKS is supported by "Tamkeen under the NYU Abu Dhabi Research Institute grant CG008 and ASPIRE Abu Dhabi under Project AARE20-336".VKS would like to thank P. Sulkowski, Q. Chen, Hisham Sati and Urs Schreiber for the helpful discussion.PR would like to thank SERB (MATRICS) MTR/2019/000956 funding, which enabled her to visit University of Warsaw and present these results.PR would also like to acknowledge the ICTP's Associate programme, which helped her to complete this project during the visit as a senior associate.SC and PR would like to thank all the speakers and the organisers of the Learning workshop on BPS states and 3-manifolds for discussions and interactions on 'knotquiver' correspondence.BPM acknowledges the research grant for faculty under IoE Scheme (Number 6031) of Ba-naras Hindu University.AD would like to thank UGC for the research fellowship.P. R. would like to acknowledge the SPARC/2019-2020/ P2116/ project funding Appendix: Quiver matrix For clarity, we present the steps in obtaining the quiver representation for a knot 8 3 in this Appendix.The Alexander polynomial in variable X is ∆ 83 (X) = (1−4X) .Such a form implied that we could perform the reverse MMR method discussed [13] for ∆ K (X) = 1 − pX with p being integer.The r-colored Jones polynomials take the following interesting form: By comparing these polynomials with known r-colored Jones polynomials, we have successfully determined the unknown parameters a ij , b i , c i .The exact expression is Using the q-binomial and q-Pochhammer identities discussed in Ref. [1], we could rewrite the r-colored Jones polynomial as J r (8 3 ; q) = r,k,α (−1) k2+k4+k6+α8 6 i=0 (q 2 ; q 2 ) k8−i−α8−i−k8−i−1+α8−i−1 (q 2 ; q 2 ) r (q 2 ; q 2 ) r−k8 6 i=0 (q 2 ; q 2 ) α8−i−α8−i−1 (q 2 ; q 2 ) k1−α1 1 (q; q) α1 q (3k2+2rk2+k 2 (q; q) r q C 8 3 (i,j) didj 17 i=1 (q 2 ; q 2 ) di q (−2d2−d3−4d4−3d5−d6) q (d8+2d9+d11−2d12−d13+d14+2d15+3d16+4d17) .
We can read off the quiver matrix elements C 83 from the above expression.See sectionII.1,where we have presented the matrix.The explicit matrix elements for C K(3,−2) and C K(3,−3) are presented below: .

II. 1 .
Colored HOMFLY-PT polynomials for a class of Double twist knots K(p, −m) III. KNOT-QUIVER CORRESPONDENCE OF DOUBLE TWIST KNOTS K(p, −m)