Braid monodromy, orderings and transverse invariants

Plamenevskaya, Olga

doi:10.2140/agt.2018.18.3691

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Braid monodromy, orderings and transverse invariants

Olga Plamenevskaya

Algebraic & Geometric Topology 18 (2018) 3691–3718

DOI: 10.2140/agt.2018.18.3691

Abstract

A closed braid $β$ naturally gives rise to a transverse link $K$ in the standard contact $3$ –space. We study the effect of the dynamical properties of the monodromy of $β$ , such as right-veering, on the contact-topological properties of $K$ and the values of transverse invariants in Heegaard Floer and Khovanov homologies. Using grid diagrams and the structure of Dehornoy’s braid ordering, we show that $\hat{θ} (K) \in \hat{HFK} (m (K))$ is nonzero whenever $β$ has fractional Dehn twist coefficient $C > 1$ . (For a $3$ –braid, we get a sharp result: $\hat{θ} \neq 0$ if and only if the braid is right-veering.)

Keywords

Heegaard Floer transverse invariants, right-veering, fractional Dehn twist coefficient, grid diagrams

Mathematical Subject Classification 2010

Primary: 57M27, 57R17, 57R58

References

Publication

Received: 21 March 2018

Revised: 8 June 2018

Accepted: 18 June 2018

Published: 18 October 2018

Authors

Olga Plamenevskaya
Department of Mathematics Stony Brook University Stony Brook, NY United States

Volume 18, issue 6 (2018)