A note on HOMFLY polynomial of positive braid links

Abstract

For a positive braid link, a link represented as a closed positive braid, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum Euler characteristics, the number of split factors, and the number of prime factors. Our results give improvements of known results for the Conway and the Jones polynomial of positive braid links. In Appendix, we present a simpler proof of theorem of Cromwell, a positive braid diagram represents a composite link if and only if the diagram is composite.