An annular refinement of the transverse element in Khovanov homology

Hubbard, Diana¹ ; Saltz, Adam¹

¹Department of Mathematics, Boston College, Maloney Hall, Fifth Floor, Chestnut Hill, MA 02467-3806, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 2305-2324

Résumé

We construct a braid conjugacy class invariant κ by refining Plamenevskaya’s transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using κ we construct an obstruction to negative destabilization (stronger than ψ) and a solution to the word problem in braid groups. Also, κ 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.

Détail

DOI : 10.2140/agt.2016.16.2305

Classification : 20F36, 57M25, 57M27, 57R17
Keywords: Khovanov homology, transverse knot, invariant, braids

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