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Colored HOMFLY polynomials of knots presented as double fat diagrams. (English) Zbl 1388.57010
Summary: 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 J. Gu and H. Jockers [Commun. Math. Phys. 338, No. 1, 393–456 (2015; Zbl 1328.81193)], 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.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57R56 Topological quantum field theories (aspects of differential topology) 81T45 Topological field theories in quantum mechanics
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