On the transverse invariant for bindings of open books

Abstract

Consider a transverse knot which is the binding of an open book for the ambient contact manifold. In this paper, we show that the transverse invariants defined by Lisca, Ozsvath, Stipsicz, and Szabo (LOSS) are nonvanishing for such transverse knots. This is true regardless of whether or not the ambient contact structure is tight. We also prove a vanishing theorem for LOSS's Legendrian and transverse invariants. As a corollary, we show that if (T,π) is an open book with connected binding, then the complement of T has no Giroux torsion.

18 pages, 8 figures. The new version contains some minor updates and typo corrections

References in corpus (6)

• Floer homology and knot complements
• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Contact structures, sutured Floer homology and TQFT
• The vanishing of the contact invariant in the presence of torsion
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• On the contact class in Heegaard Floer homology

Cited by in corpus (4)

• On invariants for Legendrian knots
• Torsion and Open Book Decompositions
• On knots in overtwisted contact structures
• Full Lutz twist along the binding of an open book