On the Transverse Invariant for Bindings of Open Books

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. Ozsvath, A. I. Stipsicz & Z. Szabo, 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.