On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants

TL;DR

This paper explores specializations of the colored HOMFLY-PT polynomial to unify various link invariants, including multivariable Alexander and Kashaev's invariants, revealing new connections in knot theory.

Contribution

It demonstrates that a broad family of multivariable link invariants from super-modules encompasses known invariants and proposes a conjecture linking them to other generalized invariants.

Findings

Multivariable link invariants include the Alexander polynomial and Kashaev's invariants.

Specializations of the invariants unify different knot invariants.

Conjecture that these invariants also relate to generalized Alexander invariants.

Abstract

We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.