Meridian twisting of closed braids and the Homfly polynomial

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 $\widehat{\beta {\Delta }^2}$.

arXiv comment14 pages

References in corpus (2)

• Matrix factorizations and link homology
• The Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond

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• A full-twisting formula for the HOMFLY polynomial
• New classes of minimal knot diagrams