On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions

Many knots and links in S 3 can be drawn as gluing of three manifolds with one or more four-punctured S 2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four -strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

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“…The most structured expansion of this type, known so far is the Hurwitz-style formula [84][85][86] for reduced colored HOMFLY:…”

Section: Quasiclassical Expansion and Vassiliev Invariantsmentioning

confidence: 99%

Colored HOMFLY polynomials of knots presented as double fat diagrams

Миронов

Морозов

Ramadevi

et al. 2015

J. High Energ. Phys.

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“…Hurwitz τ -function [5][6][7][8] is a new important subject of theoretical physics, which seems relevant to description of non-perturbative phenomena beyond 2d conformal field theory, actually beginning from the 3d Chern-Simons and knot theory, see [9][10][11]. In general, Hurwitz τ -functions do not belong [7,8] to a narrower well-studied class of KP/Toda τ -functions, i.e.…”

Section: Introductionmentioning

confidence: 99%

On KP-integrable Hurwitz functions

Alexandrov

Миронов

Морозов

et al. 2014

J. High Energ. Phys.

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101

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There is now a renewed interest [1]-[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

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“…There is an alternative genus expansion [106][107][108][109] known as Hurwitz-Fourier transform in variable h:…”

Section: Jhep08(2017)139mentioning

confidence: 99%

“…This Hurwitz version of the Fourier transform in the color index R, (1.8) converts the set of colored HOMFLY polynomials into a collection of generalized special polynomials σ K g|∆ (A) [106][107][108][109]. They enter (1.8) through…”

Section: Jhep08(2017)139mentioning

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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On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions

Cited by 45 publications

References 75 publications

Colored HOMFLY polynomials of knots presented as double fat diagrams

Colored HOMFLY polynomials of knots presented as double fat diagrams

On KP-integrable Hurwitz functions

Checks of integrality properties in topological strings

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