HOMFLY polynomials in representation [3, 1] for 3-strand braids

A. Mironov (National Research Nuclear University MEPhI, Kashirskoe shosse 31, Moscow, 115409, Russia; Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build. 1, Moscow, 127051, Russia; Theory Department, Lebedev Physics Institute, Leninsky prospekt 53, Moscow, 119991, Russia; ITEP, Bol. Cheremushkinskaya 25, Moscow, 117218, Russia) ; A. Morozov (National Research Nuclear University MEPhI, Kashirskoe shosse 31, Moscow, 115409, Russia; Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build. 1, Moscow, 127051, Russia; ITEP, Bol. Cheremushkinskaya 25, Moscow, 117218, Russia) ; An. Morozov (National Research Nuclear University MEPhI, Kashirskoe shosse 31, Moscow, 115409, Russia; Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build. 1, Moscow, 127051, Russia; Laboratory of Quantum Topology, Chelyabinsk State University, Bratiev Kashirinyh st. 129, Chelyabinsk, 454001, Russia) ; A. Sleptsov (National Research Nuclear University MEPhI, Kashirskoe shosse 31, Moscow, 115409, Russia; Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build. 1, Moscow, 127051, Russia; Laboratory of Quantum Topology, Chelyabinsk State University, Bratiev Kashirinyh st. 129, Chelyabinsk, 454001, Russia; ITEP, Bol. Cheremushkinskaya 25, Moscow, 117218, Russia)

This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R ⊗3 −→ Q with all possible Q , for R = [3 , 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3 , 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family ( n, −1 | 1 , −1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

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      "source": "Springer/SISSA", 
      "value": "This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R \u22973 \u2212\u2192 Q with all possible Q , for R = [3 , 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3 , 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family ( n, \u22121 | 1 , \u22121) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations."
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Published on:
21 September 2016
Publisher:
Springer/SISSA
Published in:
Journal of High Energy Physics (2016)

DOI:
https://doi.org/10.1007/JHEP09(2016)134
arXiv:
1605.02313
Copyrights:
The Author(s)
Licence:
CC-BY-4.0

Fulltext files: