Combinatorial knot contact homology and transverse knots

TL;DR

This paper introduces a combinatorial approach to transverse homology, an invariant of transverse knots extending knot contact homology, capable of distinguishing knots beyond existing invariants, and includes computations demonstrating its effectiveness.

Contribution

It develops a new combinatorial framework for transverse homology, extending knot contact homology, and provides computational evidence of its ability to distinguish complex transverse knots.

Findings

Transverse homology can distinguish knots that other invariants cannot.

The theory includes a three-variable polynomial related to the A-polynomial.

Computations show the invariant's effectiveness in knot classification.

Abstract

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots, including knots that cannot be distinguished by the Heegaard Floer transverse invariants or other previous invariants.