Heegaard Floer invariants of Legendrian knots in contact three--manifolds

TL;DR

This paper introduces new invariants for Legendrian and transverse knots in contact 3-manifolds using knot Floer homology, providing tools to distinguish knot types and analyze their properties.

Contribution

It defines and computes Heegaard Floer invariants for Legendrian and transverse knots, demonstrating their non-vanishing and applications in identifying transversely non-simple knots.

Findings

Invariants do not vanish for certain non-loose knots in overtwisted spheres.

Invariants help distinguish transversely non-simple knot types.

Application to many overtwisted contact 3-manifolds.

Abstract

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.