HOMFLY-PT homology for general link diagrams and braidlike isotopy

Abstract

Khovanov and Rozansky’s categorification of the homfly-pt polynomial is invariant under braidlike isotopies for any general 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, $P^b\left(D\right)$, which can be used to detect nonbraidlike isotopies. Finally, we will use $P^b\left(D\right)$ to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.