Sur le calcul d'invariants et l'engendrement des noeuds transverses dans les variétés de contact de dimension trois

Abstract : We study the properties of classical and advanced invariants for transverse knots in three-dimensional contact manifolds. In a given isotopy class, we can construct infinitely many different Legendrian knots. Colin, Giroux and Honda have proved however that, in the standard contact three-dimensional sphere, if we fix the Turston-Bennequin-invariant and the knot isotopy class, the Legendrian knots are finite. We investigate a transverse version of this result. In a first part, we show that the transverse finiteness conjecture can be reduced by push-off to the study of the finiteness of Legendrian knots which can't be destabilized. In a second part, we prove that all the Legendrian knot classes in a knot isotopy class can be obtained from a finite set of Legendrien knots with Lutz modifications on a finite number of torus. Then we explain how this generation result can be used to build a bypass in specific simple cases. In the last part, we study the cylindrical contact homology of a transverse knot by putting it in the binding of an open book decomposition with pseudo-Anosov monodromy. Then we prove that in case where the fractional Dehn twist coefficient is large enough, this contact homology has exponential growth.
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Submitted on : Tuesday, October 25, 2016 - 3:57:09 PM
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Thomas Guyard. Sur le calcul d'invariants et l'engendrement des noeuds transverses dans les variétés de contact de dimension trois . Géométrie symplectique [math.SG]. Université de Nantes, 2015. Français. ⟨tel-01387461⟩

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