Vassiliev Invariants of Virtual Legendrian Knots

TL;DR

This paper develops a theory of virtual Legendrian knots, establishing that their Vassiliev invariants are isomorphic to those of virtual framed knots, highlighting limitations in distinguishing certain virtual Legendrian knots.

Contribution

It introduces the concept of virtual Legendrian knots and proves the isomorphism of their Vassiliev invariants with those of virtual framed knots.

Findings

Vassiliev invariants of virtual Legendrian and framed knots are isomorphic.

Vassiliev invariants cannot distinguish certain isotopic virtual Legendrian knots.

The theory extends classical Legendrian knot invariants to the virtual setting.

Abstract

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.