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Transverse invariants, braids, and right-veering

Banff International Research Station for Mathematical Innovation and Discovery

Transverse links in $S^3$ can be described via braids. We will show that the "direction and amount of twisting" of such a braid determine, in many cases, whether the (hat-version of) Heegaard Floer transverse invariant of the corresponding link vanishes or not. In particular, we prove that for 3-braids, the Heegaard Floer transverse invariant is non-zero if and only if the braid is right-veering. For higher-order braids, a fractional Dehn twist coefficient greater than 1 implies non-vanishing of the invariant. This result parallels a well-known result of Honda-Kazez-Matic for open books: if an open book with connected binding has FDTC > 1, then the Heegaard Floer contact invariant is non-zero. Interestingly, the open books result uses taut foliations and symplectic fillings (there is no direct proof) whereas our result for braids follows from the combinatorial structure of Dehornoy's braid orderings and an examination of grid diagrams.

Mathematics; Manifolds and cell complexes; Global analysis, analysis on manifolds; Low dimensional topology

video/mp4

Author affiliation: State University of New York at Stony Brook

2016-09-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59220

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/