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*S² that are isotopic as virtual Legendrian knots must be Legendrian isotopic in ST*S².
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm969-10-2015,
author = {Vladimir Chernov and Rustam Sadykov},
title = {Virtual Legendrian isotopy},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {127-137},
zbl = {06602785},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm969-10-2015}
}
Vladimir Chernov; Rustam Sadykov. Virtual Legendrian isotopy. Fundamenta Mathematicae, Tome 233 (2016) pp. 127-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm969-10-2015/