Fundamental group of sextics of torus type

Abstract

We show that the fundamental group of the complement of any irreducible tame torus sextics in $\bf P^2$ is isomorphic to $\bf Z_2*\bf Z_3$ except one class. The exceptional class has the configuration of the singularities $\{C_{3,9},3A_2\}$ and the fundamental group is bigger than $\bf Z_2*\bf Z_3$. In fact, the Alexander polynomial is given by $(t^2-t+1)^2$. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves.