The HOMFLY polynomial of links in closed braid form

摘要

We provide a new proof of Jaeger's formula expressing the HOMFLY polynomial of a link presented in closed braid form, replacing the original representation theoretic proof with an easy combinatorial and geometric argument. Using new variants of Jaeger's result we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until now this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered. (C) 2018 Elsevier B.V. All rights reserved.