Khovanov-Rozansky homology and the braid index of a knot

TL;DR

This paper demonstrates that Khovanov-Rozansky homology can detect the braid index of knots where traditional inequalities fail, revealing new insights into knot invariants.

Contribution

It introduces the KR-MFW inequality based on Khovanov-Rozansky homology and shows it can detect braid indices missed by previous inequalities.

Findings

Existence of knots where MFW inequality fails but KR-MFW detects braid index

KR-MFW inequality detects braid index in infinitely many knots

Examples showing limitations of previous inequalities

Abstract

We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fails to detect the braid indices.