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July 2011 On the Transverse Invariant for Bindings of Open Books
David Shea Vela–Vick
J. Differential Geom. 88(3): 533-552 (July 2011). DOI: 10.4310/jdg/1321366359

Abstract

Let $T \subset (Y,\xi)$ be a transverse knot which is the binding of some open book, $(T,\pi )$, for the ambient contact manifold $(Y,\xi)$. In this paper, we show that the transverse invariant $\hat{\mathscr{T}(T) \in \widehat{HFK}(−Y,K)$, defined in P. Lisca, P. Ozsváth, A. I. Stipsicz & Z. Szabó, "Heegaard Floer invariants of Legendrian knots in contact three-manifolds," J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1307–1363, MR 2557137, Zbl pre05641376, is nonvanishing for such transverse knots. This is true regardless of whether or not $\xi$ is tight. We also prove a vanishing theorem for the invariants $\mathscr{L}$ and $\mathscr{T}$. As a corollary, we show that if $(T,\pi )$ is an open book with connected binding, then the complement of $T$ has no Giroux torsion.

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David Shea Vela–Vick. "On the Transverse Invariant for Bindings of Open Books." J. Differential Geom. 88 (3) 533 - 552, July 2011. https://doi.org/10.4310/jdg/1321366359

Information

Published: July 2011
First available in Project Euclid: 15 November 2011

zbMATH: 1239.53102
MathSciNet: MR2844442
Digital Object Identifier: 10.4310/jdg/1321366359

Rights: Copyright © 2011 Lehigh University

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Vol.88 • No. 3 • July 2011
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