Virtual crossings and a filtration of the triply graded homology of a   link diagram
  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. 
Michael Abel and Lev Rozansky
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