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Virtual Legendrian isotopy

Volume 234 / 2016

Vladimir Chernov, Rustam Sadykov Fundamenta Mathematicae 234 (2016), 127-137 MSC: Primary 53D10, 57R17; Secondry 57M27. DOI: 10.4064/fm969-10-2015 Published online: 24 February 2016


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 $ST^*S^2.$


  • Vladimir ChernovDepartment of Mathematics
    Dartmouth College
    6188 Kemeny Hall
    Hanover, NH 03755, U.S.A.
  • Rustam SadykovDepartment of Mathematics
    Kansas State University
    138 Cardwell Hall
    Manhattan, KS 66506, U.S.A.

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