A Note on the Jones Polynomials of 3-Braid Links

The braid group on $ n $ strands plays a central role in knot theory and lowdimensional topology.3-braids were classified, up to conjugacy, into normal forms. Basingon Burau’s representation of the braid group, Birman introduced a simple way tocalculatethe Jones polynomial of closed 3-braids. We use Birman’s formula to study thestructure 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 isdetermined 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}) $, whichare 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.