Legendrian contact homology in the product of a punctured Riemann surface and the real line

TL;DR

This paper provides a combinatorial approach to Legendrian contact homology for knots in the product of a punctured Riemann surface and the real line, revealing the existence of multiple Legendrian knots within the same smooth isotopy class.

Contribution

It introduces a combinatorial description of the Legendrian differential graded algebra in this setting and demonstrates the existence of multiple Legendrian knots with the same homology class but different Legendrian isotopy classes.

Findings

Combinatorial description of Legendrian DGA in PxR.

Existence of multiple Legendrian knots in the same homology class.

Knots are smoothly isotopic but Legendrian distinct.

Abstract

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.