## Journal of Differential Geometry

### On the Transverse Invariant for Bindings of Open Books

David Shea Vela–Vick

#### 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

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