On Alexander polynomials of torus curves

Abstract

Let $p$ and $q$ be integers such that $p>q\ge 2$ and $q$ divides $p$. Let $\varphi \left(q\right)$ be the Euler number of $q$. We exhibit a Zariski $\varphi \left(q\right)$-ple, distinguished by the Alexander polynomial, whose curves are tame torus curves of type $(p,q)$, with $q$ smooth irreducible components of degree $p$,and one single singular point topologically equivalent to the Brieskorn-Pham singularity $v^q+u^{q{p}^2}=0$.