

Geometric Topology Seminar
September 30, 2014 15:00, Moscow, Lomonosov Moscow State University, Mech.math. Dept. 1226






On knots transverse to a vector field
V. V. Chernov^{} 
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Abstract:
We classify knots in a 3manifold M that are transverse to a nowhere zero vector field V up to the corresponding isotopy relation. When V is the coorienting vector field of a contact structure, these knots are the same as pseudoLegendrian knots, which were introduced by Benedetti and Petronio. We show that two loose Legendrian knots with the same overtwisted disk in their complement are Legendrian isotopic if and only if they are pseudoLegendrian isotopic.
Vtransverse knots are naturally framed. We show that each framed isotopy class corresponds to infinitely many Vtransverse isotopy classes whose elements are pairwise distinct up to Vtransverse homotopy, provided that one of the following conditions holds: V is a coorienting vector field of a tight contact structure; the manifold M is irreducible and atoroidal; or, the Euler class of a 2dimensional bundle orthogonal to V is a torsion class.
We also give examples of infinite sets of distinct Vtransverse isotopy classes whose representatives are all Vtransverse homotopic and framed isotopic.
Language: English

