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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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