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

Abstract

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in P × R, where P is a punctured Riemann surface. As an application we show that, for any integer k and any homology class h ∈ H1(P × R), 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.

Contact Manifolds with Flexible Fillings

“…Higher dimensional examples were found in [25] by Ekholm, Etnyre, and Sullivan who produced infinitely many non-isotopic Legendrians spheres and tori in P 2n × R in the same formal class. However these Legendrians are all nullhomologous in P 2n × R. Björklund [6] showed that if Σ is a closed surface, there are arbitrarily many 1-dimensional Legendrians in Σ × R that are formally isotopic representing any class in H 1 (Σ × R). The following analog of Theorem 1.9 generalizes these results.…”