Virtual Legendrian isotopy

Vladimir Chernov¹ ; Rustam Sadykov²

¹ Department of Mathematics Dartmouth College 6188 Kemeny Hall Hanover, NH 03755, U.S.A.
² Department of Mathematics Kansas State University 138 Cardwell Hall Manhattan, KS 66506, U.S.A.

Fundamenta Mathematicae, Tome 234 (2016) no. 2, pp. 127-137

Résumé

An elementary stabilization of a Legendrian knot $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the knot $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. A class of virtual Legendrian isotopy is called a virtual Legendrian knot. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian knots and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and the first author: two Legendrian knots in $ST^*S^2$ that are isotopic as virtual Legendrian knots must be Legendrian isotopic in $ST^*S^2.$

Détail

DOI : 10.4064/fm969-10-2015

Mots-clés : elementary stabilization legendrian knot spherical cotangent bundle *m surface surgery results attaching handle along discs away image projection knot virtual legendrian isotopy composition stabilizations destabilizations legendrian isotopies class virtual legendrian isotopy called virtual legendrian knot contrast legendrian knots virtual legendrian knots enjoy property there bijective correspondence between virtual legendrian knots equivalence classes gauss diagrams study virtual legendrian knots every class contains unique irreducible representative particular get solution following conjecture cahn levi first author legendrian knots *s isotopic virtual legendrian knots legendrian isotopic *s