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On the Hecke algebras and the colored HOMFLY polynomial

Xiao-Song Lin, Hao Zheng

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-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

On the Hecke algebras and the colored HOMFLY polynomial

Abstract

The colored HOMFLY polynomial is the quantum invariant of oriented links in associated with irreducible representations of the quantum group . 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-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

Paper Structure

This paper contains 6 sections, 10 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Link invariants from quantum groups
  3. Centralizer algebra and cabling-projection rule
  4. Hecke algebras and colored HOMFLY polynomial
  5. Torus links
  6. On the Labastida-Mariño-Vafa conjecture

Key Result

Lemma 3.1

For every $x \in \mathcal{C}_n(V)$ we have

Theorems & Definitions (29)

  • Example 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Theorem 4.1: Aiston-Morton AM
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • ...and 19 more