On the topological type of a set of plane valuations with symmetries

TL;DR

This paper demonstrates that the topological type of a link formed by plane curve singularities with symmetries can be determined from a specialized Alexander polynomial, linking algebraic invariants to topological classification.

Contribution

It establishes a novel connection between Alexander polynomials and the topological type of symmetric plane curve links, extending to divisorial valuations via Poincaré series.

Findings

The Alexander polynomial $ ilde{ riangle}^L$ uniquely determines the link's topological type.

An analogous result holds for plane divisorial valuations using Poincaré series.

The work generalizes classical invariants to symmetric configurations with group actions.

Abstract

Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r\vert G\vert$ variables of the algebraic link $(\bigcup\limits_{i=1}^{r}\bigcup\limits_{a\in G}a C_i )\cap S^3_{\varepsilon}$ and let $\zeta(t_1,\ldots, t_r) = \Delta^{L}(t_1,\ldots,t_1,t_2,\ldots,t_2, \ldots,t_r,\ldots,t_r)$ with $\vert G\vert$ identical variables in each group. (If $r=1$, $\zeta(t)$ is the monodromy zeta function of the function germ $\prod\limits_{a\in G} a^*f$, where $f=0$ is an equation defining the curve $C_1$.) We prove that $\zeta(t_1,\ldots, t_r)$ determines the topological type of the link $L$. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincar\'e series of a set of valuations.