SU(N) quantum Racah coefficients & non-torus links

TL;DR

This paper derives identities for SU(N) quantum Racah coefficients using topological methods, enabling the computation of knot and link polynomials for non-torus links, which aids in testing conjectures in topological string theory.

Contribution

It introduces new identities for SU(N) Racah coefficients derived from topological arguments, facilitating calculations for non-torus knots and links.

Findings

Derived identities for SU(N) Racah coefficients.

Computed polynomials for certain non-torus knots and links.

Provided tools for verifying topological string theory conjectures.

Abstract

It is well-known that the SU(2) quantum Racah coefficients or the Wigner $6j$ symbols have a closed form expression which enables the evaluation of any knot or link polynomials in SU(2) Chern-Simons field theory. Using isotopy equivalence of SU(N) Chern-Simons functional integrals over three balls with one or more $S^2$ boundaries with punctures, we obtain identities to be satisfied by the SU(N) quantum Racah coefficients. This enables evaluation of the coefficients for a class of SU(N) representations. Using these coefficients, we can compute the polynomials for some non-torus knots and two-component links. These results are useful for verifying conjectures in topological string theory.