On the Transverse Invariant for Bindings of Open Books

David Shea Vela–Vick
2011 Journal of differential geometry  
Let T ⊂ (Y, ξ) be a transverse knot which is the binding of some open book, (T, π), for the ambient contact manifold (Y, ξ). In this paper, we show that the transverse invariant b T(T ) ∈ HFK(−Y, K), defined in [LOSS08], is nonvanishing for such transverse knots. This is true regardless of whether or not ξ is tight. We also prove a vanishing theorem for the invariants L and T. As a corollary, we show that if (T, π) is an open book with connected binding, then the complement of T has no Giroux
more » ... f T has no Giroux torsion.
doi:10.4310/jdg/1321366359 fatcat:4fh5jtu7wba73ed44vvhyyj2ue