The HOMFLY Polynomial of Links in Closed Braid Form

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, 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 know 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.