Journal of Differential Geometry

On the Transverse Invariant for Bindings of Open Books

David Shea Vela–Vick

Full-text: Open access

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.

Article information

Source
J. Differential Geom., Volume 88, Number 3 (2011), 533-552.

Dates
First available in Project Euclid: 15 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1321366359

Digital Object Identifier
doi:10.4310/jdg/1321366359

Mathematical Reviews number (MathSciNet)
MR2844442

Zentralblatt MATH identifier
1239.53102

Citation

Vela–Vick, David Shea. On the Transverse Invariant for Bindings of Open Books. J. Differential Geom. 88 (2011), no. 3, 533--552. doi:10.4310/jdg/1321366359. https://projecteuclid.org/euclid.jdg/1321366359


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