Abstract
Many knots and links in S3 can be drawn as gluing of three manifolds with one or more four-punctured S2 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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Mironov, A., Morozov, A., Morozov, A. et al. Colored HOMFLY polynomials of knots presented as double fat diagrams. J. High Energ. Phys. 2015, 109 (2015). https://doi.org/10.1007/JHEP07(2015)109
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DOI: https://doi.org/10.1007/JHEP07(2015)109