Two-bridge links and stable maps into the plane

TL;DR

This paper presents a visual method for constructing stable maps from the 3-sphere to the plane with specific properties related to two-bridge links, and calculates their complexities.

Contribution

It introduces a new visual construction technique for stable maps with prescribed link properties and computes their complexities for certain two-bridge link exteriors.

Findings

Constructed stable maps with prescribed two-bridge links

Determined stable map complexities for specific link exteriors

Identified types of fibers containing two indefinite fold points

Abstract

We give a visual construction of stable maps from the $3$-sphere into the real plane enjoying the following properties; the set of definite fold points coincides with a given two-bridge link and the map only admits certain types of fibers containing two indefinite fold points. As a corollary, we determine the stable map complexities defined by Koda and Ishikawa for some two-bridge link exteriors.