Algebraic & Geometric Topology

An annular refinement of the transverse element in Khovanov homology

Diana Hubbard and Adam Saltz

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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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2305-2324.

Dates
Received: 4 August 2015
Revised: 11 November 2015
Accepted: 4 December 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895915

Digital Object Identifier
doi:10.2140/agt.2016.16.2305

Mathematical Reviews number (MathSciNet)
MR3546466

Zentralblatt MATH identifier
1366.57004

Subjects
Primary: 20F36: Braid groups; Artin groups 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57R17: Symplectic and contact topology

Keywords
Khovanov homology transverse knot invariant braids

Citation

Hubbard, Diana; Saltz, Adam. An annular refinement of the transverse element in Khovanov homology. Algebr. Geom. Topol. 16 (2016), no. 4, 2305--2324. doi:10.2140/agt.2016.16.2305. https://projecteuclid.org/euclid.agt/1511895915


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