A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

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.

Section: Sign Dependence On Knotsmentioning

confidence: 53%

Section: Jhep07(2015)109mentioning

confidence: 86%

Section: Jhep07(2015)109mentioning

confidence: 99%

Section: Xmentioning

confidence: 99%

See 2 more Smart Citations

Colored HOMFLY polynomials of knots presented as double fat diagrams

Миронов

Морозов

Ramadevi

et al. 2015

“…which is always correct for the Racah matrices [153] and follows from triviality of two unlinked unknots [29].…”

Section: Jhep08(2017)139mentioning

confidence: 90%

Checks of integrality properties in topological strings

Mironov¹,

Морозов²,

Морозов³

et al. 2017

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.

Colored HOMFLY polynomials for the pretzel knots and links

Миронов

Морозов

Слепцов

2015

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.