Closures of $3$-braids and detection

TL;DR

This paper presents new results on detecting specific links using various homology theories, focusing on closures of 3-braids and introducing bounds on homology ranks to identify complex link patterns.

Contribution

It introduces novel detection results for links via link Floer, Khovanov, and annular Khovanov homologies, including the Mazur link and pattern detection, with new bounds on homology ranks.

Findings

Link Floer homology detects the Mazur link.

Annular Khovanov homology detects the Mazur pattern.

Khovanov homology detects links L6a2 and L9n15.

Abstract

We give some new link detection results for link Floer homology, Khovanov homology and annular Khovanov homology. The links we detect arise via different closure operations on $3$-braids. Examples of our results include that link Floer homology detects the Mazur link, that annular Khovanov homology detects the Mazur pattern, and that Khovanov homology detects L6a2 and L9n15. The Mazur pattern detection result depends on a new bound on the rank of the annular Khovanov homology of certain links.