Closures of positive braids and the Morton-Franks-Williams inequality

TL;DR

This paper investigates the Morton-Franks-Williams inequality for positive braid closures, proving the basis property of simple braids in the Hecke algebra and introducing a new technique using resolution trees to analyze the Homflypt polynomial.

Contribution

It provides a simple proof that simple braids form an orthonormal basis and introduces resolution trees to study the Homflypt polynomial for positive braid closures.

Findings

Simple braids form an orthonormal basis for the Hecke algebra.

Resolution trees characterize when the Morton-Franks-Williams inequality is strict.

Explicit classification of positive braids on three strands with braid index three.

Abstract

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by K\'alm\'an, who first obtained this result by using an interesting connection with Contact Topology. We also introduce a new technique to study the Homflypt polynomial for closures of positive braids, namely resolution trees whose leaves are simple braids. In terms of these simple resolution trees, we characterize closed positive braids for which the Morton-Franks-Williams inequality is strict. In particular, we determine explicitly the positive braid words on three strands whose closures have braid index three.