Combinatorial computation of the motivic Poincare series

TL;DR

This paper presents an explicit algorithm for computing the motivic Poincare series of plane curve singularities using embedded resolution, linking it to Alexander polynomials and Heegaard-Floer homologies, with numerous examples.

Contribution

It introduces a new algorithm for calculating the motivic Poincare series based on embedded resolution, connecting it to known invariants like Alexander polynomials and knot homologies.

Findings

The algorithm produces a rational function depending on parameter q.

At q=1, the series coincides with the Alexander polynomial.

Explicit examples demonstrate the method's effectiveness.

Abstract

We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve. The result is a rational function depending of the parameter q, at q=1 it coincides with the Alexander polynomial of the corresponding link. For irreducible curves we relate this invariant to the Heegard-Floer knot homologies constructed by P. Ozsvath and Z. Szabo. Many explicit examples are considered.