Virtual crossings and a filtration of the triply graded homology of a link diagram

TL;DR

This paper introduces a new filtration of triply graded link homology using virtual crossings, demonstrating invariance under Reidemeister and Markov moves and analyzing the effects of the third move on the filtration.

Contribution

It extends the filtration of Soergel bimodules to Rouquier complexes and proves invariance properties of the filtered homology under Reidemeister and Markov moves.

Findings

Filtration of triply graded link homology is invariant under Markov moves.

Invariance of complexes under the second Reidemeister move up to filtered homotopy.

Homotopy equivalence under the third Reidemeister move affects filtration by at most two units.

Abstract

A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier's complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move, and the filtration of the triply graded link diagram homology, constructed by Khovanov through the application of the Hochschild homology, is invariant under Markov moves. We also prove that the homotopy equivalence of the complexes of braid words related by the third Reidemeister move violates filtration by at most two units.