An annular refinement of the transverse element in Khovanov homology

Hubbard, Diana; Saltz, Adam

doi:10.2140/agt.2016.16.2305

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An annular refinement of the transverse element in Khovanov homology

Diana Hubbard and Adam Saltz

Algebraic & Geometric Topology 16 (2016) 2305–2324

DOI: 10.2140/agt.2016.16.2305

Abstract

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.

Keywords

Khovanov homology, transverse knot, invariant, braids

Mathematical Subject Classification 2010

Primary: 20F36, 57M25, 57M27, 57R17

References

Publication

Received: 4 August 2015

Revised: 11 November 2015

Accepted: 4 December 2015

Published: 12 September 2016

Authors

Diana Hubbard
Department of Mathematics Boston College Maloney Hall, Fifth Floor Chestnut Hill, MA 02467-3806 United States
http://sites.google.com/site/dianadhubbard/
Adam Saltz
Department of Mathematics Boston College Maloney Hall, Fifth Floor Chestnut Hill, MA 02467-3806 United States
http://www2.bc.edu/adam-r-saltz

Volume 16, issue 4 (2016)