An instanton take on some knot detection results

TL;DR

This paper proves that certain knot homologies uniquely identify specific knots, using instanton Floer homology techniques inspired by Tom Mrowka's work, extending previous results with new methods.

Contribution

It introduces a novel approach employing instanton Floer homology to establish knot detection results previously shown with knot Floer homology.

Findings

Khovanov homology detects the figure eight knot and cinquefoils.

HOMFLY homology detects the 5_2 knot and certain pretzel knots.

The proofs adapt existing methods to the instanton Floer homology framework.

Abstract

We give new proofs that Khovanov homology detects the figure eight knot and the cinquefoils, and that HOMFLY homology detects $5_2$ and each of the $P(-3,3,2n+1)$ pretzel knots. For all but the figure eight these mostly follow the same lines as in previous work. The key difference is that in honor of Tom Mrowka's 60th birthday, the arguments here use instanton Floer homology rather than knot Floer homology.