Full twists factorization formula for double number of strings

Abstract

A formula for factorizations of the full twist in the braid group $Br_{2m}$ depending on any four factorizations of the full twist in $Br_{m}$ is given. Applying this formula, a symplectic 4-manifold $X$ and two isotopic generic coverings $f_i :X\to \mathbb C\mathbb P^2$, branched, respectively, along cuspidal Hurwitz curves $\bar H_i\subset \mathbb C\mathbb P^2$ (without negative nodes) having different braid monodromy factorization types, are constructed. The class of the fundamental groups of the complements of the affine plane Hurwitz curves is described in terms of generators and defining relations.