Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids

TL;DR

This paper derives explicit formulas for HOMFLY polynomials of 5-, 6-, and 7-strand braids using Racah coefficients for SU_q(3), enabling comprehensive computation of these knot invariants across various braid configurations.

Contribution

It provides the first explicit formulas for all 5-, 6-, and 7-strand Wilson averages in the fundamental representation, extending the understanding of HOMFLY polynomials for complex braids.

Findings

Formulas reproduce all known 5-strand knot HOMFLY polynomials with 9 crossings.

7-strand formulas are sufficient to describe all HOMFLY polynomials in the knot atlas.

The approach suggests potential universality across quantum groups for these invariants.

Abstract

Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand formulas are sufficient to reproduce all the HOMFLY polynomials from the katlas.org: they are all described at once by a simple explicit formula with a very transparent structure. Moreover, would the formulas for the relevant SU_q(3) Racah coefficients remain true for all other quantum groups, the paper provides a complete description of the fundamental HOMFLY polynomials for all braids with any number of strands.