Legendrian Contact Homology in Seifert Fibered Spaces

TL;DR

This paper introduces a new Legendrian contact homology invariant for knots in Seifert fibered spaces, accounting for orbifold points, and demonstrates its ability to distinguish knots with identical topological types.

Contribution

It defines a novel differential graded algebra for Legendrian knots in Seifert fibered spaces with transverse contact structures, incorporating orbifold points into the invariant.

Findings

Invariant distinguishes Legendrian knots with torsion homology

Applicable to knots with identical topological types in complex spaces

Shows the importance of orbifold points in contact homology

Abstract

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology is torsion and whose underlying topological knot types are isotopic; such examples exist in any sufficiently complicated contact Seifert fibered space.