Essential tori associated with links of mixed singularities

TL;DR

This paper links the analytic properties of weakly isolated mixed singularities to the topology of their links, providing explicit criteria to detect essential tori and non-hyperbolicity from polynomial data.

Contribution

It introduces a new method to determine the presence of essential tori in link complements directly from mixed polynomial properties, bridging analytic and topological aspects.

Findings

Criteria for essential tori detection are explicit and computable.

Conditions are provided to ensure links are non-hyperbolic.

The approach avoids explicit link type determination.

Abstract

We establish a direct connection between the analytic data of weakly isolated mixed singularities and the topology of their associated links. More precisely, we prove that the existence of essential tori, topological information, in the complements of links arising from weakly isolated mixed singularities can be detected directly from properties of the defining mixed polynomial, provided that it is convenient, non-degenerate and $\Gamma$-nice. Our results provide explicit and computable criteria, expressed purely in terms of the polynomial data, that determine the presence of essential tori in the link exterior. In particular, these criteria yield effective conditions ensuring that such links are non-hyperbolic. This approach provides a new method to extract topological information about link complements without requiring an explicit determination of the link type, thereby establishing a concrete bridge between the analytic structure of mixed polynomials and the geometric topology of their associated links.