Stable maps and hyperbolic links

TL;DR

This paper characterizes hyperbolic links in the 3-sphere that admit stable maps into the plane with a specific singular fiber structure, advancing understanding of link mappings and singularity configurations.

Contribution

It provides a complete characterization of hyperbolic links in the 3-sphere that admit stable maps with a single fiber having two singular points.

Findings

Characterization of hyperbolic links with specific stable map properties

Identification of stable maps with exactly one singular fiber with two points

Advancement in understanding link mappings and singularities

Abstract

A stable map of a closed orientable $3$-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links in the $3$-sphere that admit stable maps into the real plane with exactly one (connected component of a) fiber having two singular points.