On a canonical polynomial for links of elliptic singularities

TL;DR

This paper explores the relationship between the canonical polynomial and the elliptic sequence in elliptic surface singularities, revealing how the polynomial's exponents encode the elliptic structure and defining conditions for their extensions.

Contribution

It establishes a link between the exponents of the canonical polynomial and the elliptic sequence, and introduces the concept of good extensions for elliptic germs.

Findings

Exponents of the canonical polynomial determine the elliptic sequence.

A characterization of good extensions via inclusion formulas.

Compatibility of polynomial exponents with the flag structure of the elliptic sequence.

Abstract

The canonical polynomial is an important output of the multivariable topological Poincar\'e series associated with a normal surface singularity. It can be considered as a multivariable polynomial generalization of the Seiberg--Witten invariant of the link. In the case of elliptic germs, another key topological invariant was considered, the elliptic sequence, which mirrors the specific structure of the elliptic germs and guides several properties of them. In this note we study the relationship of these two objects. First of all, we describe the structure of the exponents of the canonical polynomial and prove that they determine the elliptic sequence. For the converse problem, we consider an inductive setup of elliptic germs via natural extension of their graphs and compare the corresponding sets of exponents. This leads to the definition of a good extension which can be characterized by an inclusion type formula for the corresponding canonical polynomials. This reflects in a compatible way the `flag structure' of the elliptic sequence.