Vela-Vick, David Shea On the transverse invariant for bindings of open books. (English) Zbl 1239.53102 J. Differ. Geom. 88, No. 3, 533-552 (2011). Two important theorems are proved. Namely: Theorem 1. Let \(T\subset (Y,\xi)\) be a transverse knot which can be realized as the binding of an open book \((T,\pi)\) compatible with the contact structure \(\xi\). Then the transverse invariant \(\widetilde T(T)\) is nonvanishing. Theorem 2. Let \(L\) be a Legendrian knot in a contact manifold \((Y,\xi)\). If the complement \(Y\setminus L\) contains a compact submanifold \(N\) with a convex boundary such that \(c(N,\xi|_N)= 0\) in \(SFH(-N,\Gamma)\), then the Legendrian invariant \(L(L)\) vanishes. From the second theorem the following corollary is deduced: Corollary. Let \(L\) be a Legendrian knot in a contact manifold \((Y,\xi)\). If the complement \(Y\setminus L\) has positive Giroux torsion, then the Legendrian invariant \(L(L)\) vanishes. Reviewer: Mihail Banaru (Smolensk) Cited in 1 ReviewCited in 8 Documents MSC: 53D10 Contact manifolds (general theory) 57M27 Invariants of knots and \(3\)-manifolds (MSC2010) Keywords:contact manifold; Legendrian knot; transverse knot; Legendrian invariant; transverse invariant; Giroux torsion Citations:Zbl 1232.57017 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid