Meridian twisting of closed braids and the Homfly polynomial

TL;DR

This paper establishes a precise relationship between specific coefficients of the Homfly polynomial for a closed braid and its meridian-twisted version, revealing conditions for the sharpness of degree estimates.

Contribution

It proves a new coefficient coincidence in the Homfly polynomial relating a braid and its Garside half-twist, linking degree estimates to braid twisting.

Findings

Coefficient of v^{w-n+1} in the Homfly polynomial matches a scaled coefficient in the twisted braid.

The sharpness of the Morton--Franks--Williams degree estimate is characterized by this coefficient coincidence.

The result provides a criterion for when the degree bounds in the Homfly polynomial are tight.

Abstract

Let $\beta$ be a braid on $n$ strands, with exponent sum $w$. Let $\Delta$ be the Garside half-twist braid. We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $\beta$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $\beta\Delta^2$. This coincidence implies that the lower Morton--Franks-Williams estimate for the $v$--degree of the Homfly polynomial of $\hat\beta$ is sharp if and only if the upper MFW estimate is sharp for the $v$--degree of the Homfly polynomial of $\hat{\beta\Delta^2}$.