We construct a braid conjugacy class invariant by refining Plamenevskaya’s transverse element in Khovanov homology via the annular grading. While is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using we construct an obstruction to negative destabilization (stronger than ) and a solution to the word problem in braid groups. Also, is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.
"An annular refinement of the transverse element in Khovanov homology." Algebr. Geom. Topol. 16 (4) 2305 - 2324, 2016. https://doi.org/10.2140/agt.2016.16.2305