New structures for colored HOMFLY-PT invariants

TL;DR

This paper introduces new algebraic structures for colored HOMFLY-PT invariants, proving their integrality and symmetry properties, and applying these results to refine the LMOV conjecture and related link polynomials.

Contribution

It establishes the strong integrality of normalized colored HOMFLY-PT invariants using skein theory and applies this to enhance understanding of the LMOV conjecture and link polynomial specializations.

Findings

Proved strong integrality of normalized colored HOMFLY-PT invariants.

Derived symmetric properties for full colored HOMFLY-PT invariants.

Linked the colored Alexander polynomial conjecture to the LMOV conjecture for framed knots.

Abstract

In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by H. Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then, we apply our results to refine the mathematical structures appearing in the Labastida-Mari\~no-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another application of the strong integrality, we obtain that the $q=1$ and $a=1$ specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. We find that a conjectural formula for the colored Alexander polynomial which is the $a=1$ specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for frame knot holds.