HOMFLY-PT homology for general link diagrams and braidlike isotopy

TL;DR

This paper investigates the invariance properties of HOMFLY-PT homology, revealing its limitations under certain Reidemeister moves and virtual link presentations, and introduces a deformed polynomial to detect nonbraidlike isotopies.

Contribution

The authors prove that HOMFLY-PT homology is not invariant under the Reidemeister IIb move and extend the understanding of its behavior for general link diagrams, including virtual links.

Findings

Reidemeister IIb move fails in HOMFLY-PT homology

Decategorification yields a deformed HOMFLY-PT polynomial

HOMFLY-PT homology is not invariant for virtual links

Abstract

Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in HOMFLY-PT homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of HOMFLY-PT homology for general link diagrams gives a deformed version of the HOMFLY-PT polynomial, $P^{b}(D)$, which can be used to detect nonbraidlike isotopies. Finally, we will use $P^{b}(D)$ to prove that HOMFLY-PT homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.