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On the transverse invariant for bindings of open books. (English) Zbl 1239.53102
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.

MSC:
53D10 Contact manifolds, general
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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