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.