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Maximal Thurston–Bennequin number and reducible Legendrian surgery

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  30 June 2016

Kouichi Yasui
Kouichi Yasui*
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan email kyasui@hiroshima-u.ac.jp
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Abstract

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

Keywords

4-manifoldLegendrian knotStein manifoldDehn surgerycabling conjecture

MSC classification

Primary: 57R17: Symplectic and contact topology 57R65: Surgery and handlebodies 57M25: Knots and links in $S^3$
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1899 - 1914
Copyright
© The Author 2016 

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