The homology polynomial and the burau representation for pseudo-anosov braids

The homology polynomial is an invariant for pseudo-Anosov mapping classes (Birman, 2010). We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection tothe Burau representation. Given a pseudo-Anosov braid (Îø ⁸́⁸ Bn), we determine necessary and sufficient conditions under which the homology polynomial of (Îø) is equal to the the characteristic polynomial of the image of (Îø) under the Burau representation. In particular, we build upon Band (2007) and show that the orientation cover associated to a pseudo-Anosov braid is equivalent to a quotient to the Burau cover when the measured foliations associated to (Îø) have odd-ordered singularities at each puncture and any singularity that occurs in the interior of (Dn) is even-ordered.We next construct an algorithm which allows us to determine the homology polynomial from the Burau representation for an arbitrary pseudo-Anosov braid. As an application, we show how to easily determine the homology polynomial for large family of pseudo-Anosov braids.