Virtual Legendrian Isotopy

V. Chernov; R. Sadykov

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arXiv · 1406.0875

Virtual Legendrian Isotopy

V. Chernov·R. Sadykov

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Abstract

An elementary stabilization of a Legendrian link $L$ 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 link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. 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 isotopy classes of Legendrian links 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 $ST^*S^2.$

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BibTeX RIS

V. Chernov, R. Sadykov. 2014-06-03. Virtual Legendrian Isotopy. https://arxiv.org/abs/1406.0875

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