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Two-bridge links and stable maps into the plane

Kazuhiro Ichihara, Gakuto Kato

TL;DR

This work addresses constructing stable maps from $S^{3}$ to $\mathbb{R}^{2}$ whose definite fold set equals a given two-bridge link $L$, while precisely controlling the types and counts of singular fibers, notably the $\mathrm{II}^{2}$ and $\mathrm{II}^{3}$ fibers. The authors present two explicit constructions, $f_{2}$ and $f_{3}$, for two-bridge links represented by Conway forms $C(a_{1},b_{1},\dots,a_{m},b_{m},a_{m+1})$ with all $b_{i}$ even: $f_{2}$ yields $S_{0}(f_{2})=L$ with $|\mathrm{II}^{2}(f_{2})|=2m$ and $|\mathrm{II}^{3}(f_{2})|=0$, while $f_{3}$ yields $S_{0}(f_{3})=L$ with $|\mathrm{II}^{2}(f_{3})|=0$ and $|\mathrm{II}^{3}(f_{3})|=\tfrac{1}{2}\sum_{i=1}^{m}|b_{i}|$. These constructions feed into stable-map complexity analyses, giving $smc(E(L))\le 2m$ and, for large even parameters, $smc(E(L))=2m$, with hyperbolic-volume bounds supporting the sharpness. The results connect combinatorial data from Conway forms to explicit topological and geometric properties of two-bridge link exteriors.

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.

Two-bridge links and stable maps into the plane

TL;DR

This work addresses constructing stable maps from to whose definite fold set equals a given two-bridge link , while precisely controlling the types and counts of singular fibers, notably the and fibers. The authors present two explicit constructions, and , for two-bridge links represented by Conway forms with all even: yields with and , while yields with and . These constructions feed into stable-map complexity analyses, giving and, for large even parameters, , with hyperbolic-volume bounds supporting the sharpness. The results connect combinatorial data from Conway forms to explicit topological and geometric properties of two-bridge link exteriors.

Abstract

We give a visual construction of stable maps from the -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.
Paper Structure (4 sections, 3 theorems, 5 equations, 23 figures)

This paper contains 4 sections, 3 theorems, 5 equations, 23 figures.

Table of Contents

  1. Introduction
  2. A construction of $f_{2}$
  3. A construction of $f_{3}$
  4. Stable map complexities

Key Result

Theorem 1.1

Let $L$ be a two-bridge link in $S^{3}$ represented by a Conway form $C(a_{1},b_{1},\cdots,a_{m},b_{m},a_{m+1} )$ with non-zero integers $a_{1},b_{1},\cdots,a_{m},b_{m},a_{m+1}$. If $b_{i}$ is even for all $i$, then there exists a stable map $f_{2}$ from $S^{3}$ into $\mathbb{R}^{2}$ satisfying that

Figures (23)

  • Figure 1: The types of the singular fibers.
  • Figure 2: Conway form $C(a_{1},b_{1},\cdots,a_{m},b_{m},a_{m+1})$.
  • Figure 3: The diagram $D^{\prime}$ of $L$.
  • Figure 4: The immersed curve $C^{\prime}$.
  • Figure 5: The four types of rectangular regions.
  • ...and 18 more figures

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 2.1
  • Example 3.1
  • Remark 4.1