2014/06/03 by Chernov, V., Sadykov, R.
#57R17 Secondary: 57M27 #FOS: Mathematics #Geometric Topology (math.GT) #Primary: 53D10 #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1406.0875
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^*S2 that are isotopic as virtual Legendrian knots must be Legendrian isotopic in ST^*S2.