Knot contact homology detects cabled, composite, and torus knots

TL;DR

This paper proves that knot contact homology can uniquely identify torus, cabled, and composite knots, establishing it as a powerful invariant in knot theory.

Contribution

It demonstrates that knot contact homology detects all torus knots and distinguishes cabled and composite knots from others, advancing the understanding of knot invariants.

Findings

Knot contact homology detects all torus knots.

Isomorphic knot contact homology implies the knot is cabled or composite.

Provides new tools for classifying knots using contact homology.

Abstract

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).