Filtrations on the knot contact homology of transverse knots

TL;DR

This paper introduces a new doubly filtered invariant for transverse links in R^3, based on knot contact homology DGA, which distinguishes transverse links beyond existing invariants.

Contribution

It constructs a novel doubly filtered knot contact homology invariant for transverse links, with a combinatorial formula and independence from prior invariants.

Findings

The invariant is a doubly filtered DGA derived from holomorphic disk intersections.

A combinatorial braid-based formula for the filtered DGA is provided.

The new invariant distinguishes transverse links beyond previous invariants.

Abstract

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.