Sharpness of the Morton-Franks-Williams inequality for positive knots and links

TL;DR

This paper characterizes positive diagrams that meet the Morton-Franks-Williams bound for the HOMFLY-PT polynomial, enabling easier generation of diagrams with specific properties and proposing a conjecture for strongly quasipositive links.

Contribution

It provides a combinatorial characterization of diagrams satisfying the equality in the Morton-Franks-Williams inequality, facilitating the construction of diagrams with desired invariants.

Findings

Characterization of positive diagrams meeting the inequality

Method for generating diagrams with specific invariants

Conjecture on sharpness for strongly quasipositive links

Abstract

We provide a combinatorial characterisation of positive diagrams satisfying the equality in the Morton-Franks-Williams bound for the degrees of the HOMFLY-PT polynomial. This characterisation allows generating with relative ease examples of diagrams realizing the crossing number, the braid index, and the maximal self-linking number. Besides, we suggest a conjecture concerning the sharpness of the Morton-Franks-Williams inequality for strongly quasipositive links.