Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.
Comment: 30 pages, 32 figures; (v2) the title changed, Section 1 rewritten, remarks added. arXiv admin note: text overlap with arXiv:1403.7946