On the Hecke algebras and the colored HOMFLY polynomial

TL;DR

This paper derives a new formula for the colored HOMFLY polynomial of links using Hecke algebra characters, facilitating tests of the Labastida-Mari o-Vafa conjecture and deepening the connection between quantum invariants, gauge theory, and string theory.

Contribution

It introduces a novel approach to compute the colored HOMFLY polynomial via Hecke algebra characters and Schur polynomials, simplifying calculations for torus links.

Findings

Derived a formula for colored HOMFLY polynomial in terms of Hecke algebra characters

Provided a simple formula for torus links' colored HOMFLY polynomial

Enabled testing of the Labastida-Mari o-Vafa conjecture on torus links

Abstract

The colored HOMFLY polynomial is the quantum invariant of oriented links in $S^3$ associated with irreducible representations of the quantum group $U_q(\mathrm{sl}_N)$. In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mari\~no-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.