Tangle contact homology

TL;DR

This paper extends knot contact homology to tangles, providing a new invariant that detects trivial tangles and satisfies a gluing formula, enhancing the understanding of tangle invariants in topology.

Contribution

It introduces tangle contact homology as a new invariant and proves a gluing formula, expanding the applicability of contact homology to tangles.

Findings

Tangle contact homology is an invariant under ambient isotopy.

Gluing of tangles satisfies a specific gluing formula.

Tangle contact homology detects trivial tangles.

Abstract

Knot contact homology is an ambient isotopy invariant of knots and links in $\mathbb R^3$. The purpose of this paper is to extend this definition to an ambient isotopy invariant of tangles and prove that gluing of tangles gives a gluing formula for knot contact homology. As a consequence of the gluing formula we obtain that the tangle contact homology detects triviality of tangles.