121155431
Cabling procedure for the colored HOMFLY polynomials
We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$-matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$-matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$-matrices and clarifying some conjectures formulated in previous papers.
119162645
Towards matrix model representation of HOMFLY polynomials
We investigate possibilities of generalizing the TBEM (Tierz, Brini-Eynard-Mariño) eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitian models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N = 2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in ħ = logq. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz τ-function approach to knot theory, but this remains a subject for future investigation.
119155537
Evolution method and HOMFLY polynomials for virtual knots
Following the suggestion of Alexei Morozov, Andrey Morozov and Anton Morozov, Phys. Lett. B737, 48 (2014), arXiv:1407.6319, to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel two-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.
119623409
On possible existence of HOMFLY polynomials for virtual knots
Abstract Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N = 2 to arbitrary N of the Kauffman–Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance – thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.
55778932
Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux
If a knot is represented by an -strand braid, then HOMFLY polynomial in representation is a sum over characters in all representations . Coefficients in this sum are traces of products of quantum -matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity of in . If is the fundamental representation , then is equal to the number of paths in representation graph, which lead from the fundamental vertex to the vertex . In the basis of paths the entries of the relevant -matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary -matrices consist of just and blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation , then the braid has as many as strands; have a bigger size , but only paths passing through the vertex are included into the sums over paths which define the products and traces of the relevant -matrices. In the case of , this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.
119577066
HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
A bstractExplicit answer is given for the HOMFLY polynomial of the figure eight knot 41 in arbitrary symmetric representation R = [p]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the identity for the “special” polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation (“non-commutative $ \mathcal{A} $-polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.
204007506
Proving AGT conjecture as HS duality: Extension to five dimensions
Abstract We extend the proof from Mironov et al. (2011) [25] , which interprets the AGT relation as the Hubbard–Stratonovich duality relation to the case of 5 d gauge theories. This involves an additional q -deformation. Not surprisingly, the extension turns out to be straightforward: it is enough to substitute all relevant numbers by q -numbers in all the formulas, Dotsenko–Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β = 1 , i.e. for q = t in MacDonaldʼs notation. For β ≠ 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams.
119172776
Genus expansion of HOMFLY polynomials
In the planar limit of the’ t Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern-Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagram), HR(A|q)|q=1 = (σ1(A)|R|. As a result, the (knot-dependent) Ooguri-Vafa partition function $$\sum\nolimits_R {H_{R\chi R} \left\{ {\bar pk} \right\}}$$ becomes a trivial τ -function of the Kadomtsev-Petviashvili hierarchy. We study higher-genus corrections to this formula for HR in the form of an expansion in powers of z = q − q−1. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the z-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number N of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.
118994915
Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials
Original proofs of the AGT relations with the help of the Hubbard–Stratanovich duality of the modified Dotsenko–Fateev matrix model did not work for β ≠ 1, because Nekrasov functions were not properly reproduced by Selberg–Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper, we describe a complete proof of AGT in the simple case of GL(2) (N = 2) Yang–Mills theory, i.e., the 4-point spherical conformal block of the Virasoro algebra.
117639345
Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation
An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory.
119611129
Differential hierarchy and additional grading of knot polynomials
Colored knot polynomials have a special Z-expansion in certain combinations of differentials, which depend on the representation. The expansion coefficients are functions of three variables A, q, and t and can be regarded as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of the alternative character expansion. These new variables decompose especially simply when the representation is embedded into a product of fundamental representations. The recently proposed fourth grading is seemingly a simple redefinition of these new coordinates, elegant, but in no way distinguished. If this is so, then it does not provide any new independent knot invariants, but it can instead be regarded as one more piece of evidence in support of a hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.