A NOTE ON THE JONES POLYNOMIALS OF 3-BRAID LINKS

The braid group on n strands plays a central role in knot theory and low dimensional topology. 3-braids were classified, up to conjugacy, into normal forms. Basing on Burau's representation of the braid group, Birman introduced a simple way to calculate the Jones polynomial of closed 3-braids. We use Birman's formula to study the structure of the Jones polynomial of links of braid index 3. More precisely, we show that in many cases the normal form of the 3-braid is determined by the Jones polynomial and the signature of its closure. In particular we show that alternating pretzel links P(1, c(1), c(2), c(3)), which are known to have braid index 3, cannot be represented by alternating 3-braids. Also we give some applications to the study of symmetries of 3-braid links.