Weil-Poincar\'e series and topology of collections of valuations on rational double points

TL;DR

This paper introduces the Weil-Poincaré series for collections of valuations on rational double point singularities, demonstrating that, in most cases, it determines the topology of the associated link, extending previous results related to Alexander polynomials.

Contribution

It defines the Weil-Poincaré series for valuations on rational double points and shows it generally determines the link topology, broadening the understanding of valuation invariants.

Findings

Weil-Poincaré series often determine link topology.

Explicit exceptions where the series does not determine topology.

Analogous results for divisorial valuations.

Abstract

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincar\'e sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincar\'e sphere coincides with the Poincar\'e series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the $E_8$-singularity. Here we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil-Poincar\'e series. We show that, except a few explicitly described cases the Weil-Poincar\'e series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.