Abstract
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.
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