Virtual crossings and filtrations in link homology

In 2006 Khovanov and Rozansky introduced a triply-graded link homology theory categorifying the HOMFLY-PT polynomial. Khovanov later gave an alternate construction of HOMFLY-PT homology using Rouquier's braid group action on the category of Soergel bimodules. Soergel bimodules can be filtered by submodules which are the images of virtual crossings in an action of the virtual braid group on the category of graded bimodules over polynomial rings. We conjecture that this filtration extends to HOMFLY-PT homology. We prove that the filtered version of HOMFLY-PT homology is invariant under Reidemeister I and II moves, and conjecture that Reidemeister III does the same. We show that Reidemeister III can violate filtration by at most two levels. This filtration would give a fourth grading on HOMFLY-PT homology, which has been suggested by experimental calculations in recent physics research. The use of filtrations allows us to replace proofs done by generators and relations for Soergel bimodules with more intuitive and diagrammatic proofs.