Colored HOMFLY polynomials of knots presented as double fat diagrams

A bstract 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 t...

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Bibliographic Details
Published in:The journal of high energy physics 2015-07, Vol.2015 (7), p.1-70, Article 109
Main Authors: Mironov, A., Morozov, A., Morozov, An, Ramadevi, P., Singh, Vivek Kumar
Format: Article
Language:English
Subjects:
Boundaries
Braiding
Classical and Quantum Gravitation
Elementary Particles
Evolution
Graphs
High energy physics
Knots
Physics
Physics and Astronomy
Polynomials
Pretzels
Quantum Field Theories
Quantum Field Theory
Quantum Physics
Regular Article - Theoretical Physics
Relativity Theory
Representations
String Theory
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Summary:A bstract 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.
ISSN:1029-8479
1029-8479
DOI:10.1007/JHEP07(2015)109