Colored Kauffman homology and super-A-polynomials

Nawata, Satoshi; Ramadevi, P.; Zodinmawia,

doi:10.1007/jhep01(2014)126

Cited by 27 publications

(20 citation statements)

References 126 publications

(304 reference statements)

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“…To define them, one needs "initial conditions" for the evolution, i.e. explicit knowledge of knot polynomials for a few particular values of n. Despite an extreme naiveness of the evolution method it allowed one to study certain interesting families, in particular, the important family of twist knots [39] and led to a discovery of a very important "differential structure" [40,41] of arbitrary knot polynomials, which seems related to the original ideas in [31], and led to a number of impressive advances in knot calculus, at least, for symmetric representations [42][43][44][45][46][47][48][49][50][51]. (However, attempts to generalize the matrix model (1.4) in [52,53] and to describe nonsymmetric representations in [54][55][56] are still only partly successful.…”

Section: Jhep07(2015)069mentioning

confidence: 99%

Colored HOMFLY polynomials for the pretzel knots and links

Миронов

Морозов

Слепцов

2015

J. High Energ. Phys.

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With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g + 1 two strand braids, parallel or antiparallel, and depend on g + 1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g + 1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU q (N ) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

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Section: Jhep07(2015)069mentioning

confidence: 99%

Colored HOMFLY polynomials for the pretzel knots and links

Миронов

Морозов

Слепцов

2015

J. High Energ. Phys.

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“…Since there is a double product/sum in (2) and (3), it is natural to consider their refinement, where a second t-parameter is introduced, usually called q (for knots this means going from HOMFLY to super-and hyper-polynomials [69][70][71][72][73][74][75][76][77][78][79] …”

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confidence: 99%

On factorization of generalized Macdonald polynomials

Kononov

Морозов

2016

Eur. Phys. J. C

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A remarkable feature of Schur functions-the common eigenfunctions of cut-and-join operators from W ∞ -is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U q (SL N ) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorizationon a one-(rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.Generalized Macdonald polynomials (GMPs) [1-3] play a constantly increasing role in modern studies of the 6d version of AGT relations [31][32][33][34][35][36][37][38][39][40] and spectral dualities [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [59,60]. Even the simplest questions about them are yet unanswered. In this letter we address one of them-what happens to the hook formulas for classical, quantum, and Macdonald dimensions at the level of GMPs? We find that they survive, but only partly-on a onedimensional line in the space of time variables. Lifting to the entire two-dimensional topological locus remains to be found.We begin by recalling that the Schur functions χ R { p}, depend on representation (Young diagram) R and on infinitely many time variables p k (actually, a particular χ R depends a e-mail: morozov@itep.ru only on p k with k ≤ |R| = # boxes in R). They get nicely factorized on a peculiar two-dimensional topological locus,Coarm a and coleg l are the ordinary coordinates of the box in the diagram. To keep the notation consistent throughout the text, in (1) we call the relevant parameter t, not q, from the very beginning.It is often convenient to ignore the simple overall coefficient and substitute the product formulas like (2) by a polynomial expression for a plethystic logarithm ⎛

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“…It is appropriate to mention that Marino's conjectures have been verified for some torus knots and links [124][125][126][127] and figure-eight knot [128]. Hence the main focus in this paper is to verify Marino's integrality conjectures for various arborescent knots up to 8 crossings using colored Kauffman polynomials (SO(N ) colors up to two boxes in Young diagram) and HOMFLY polynomials for mixed SU(N ) representations.…”

Section: Jhep08(2017)139mentioning

confidence: 99%

“…Our results for many knots and links can be considered as a direct continuation of the appendix from [77] and especially of appendix B from [128] for figure-eight knot where the theory is presented in detail with relevant references.…”

Section: Jhep08(2017)139mentioning

confidence: 99%

“…A natural generalization of the described correspondence is the equivalence between the SO/Sp Chern-Simons theory and the topological string theory on an orientifold of the small resolution of the conifold [133]- [128]. In this case, the Chern-Simons partition function is associated with two types of contributions: those from oriented and non-oriented strings, the former ones coming with the degree 1/2: 3…”

Section: Integrality Conjectures In the Kauffman Casementioning

confidence: 99%

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Checks of integrality properties in topological strings

Mironov¹,

Морозов²,

Морозов³

et al. 2017

J. High Energ. Phys.

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Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N ) and SO(N ) adjoint representations [1] are useful to verify Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.

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Colored Kauffman homology and super-A-polynomials

Cited by 27 publications

References 126 publications

Colored HOMFLY polynomials for the pretzel knots and links

Colored HOMFLY polynomials for the pretzel knots and links

On factorization of generalized Macdonald polynomials

Checks of integrality properties in topological strings

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