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Divides with cusps and symmetric links

Sakumi Sugawara

Abstract

A Divide with cusps is the image of a proper generic immersion from finite intervals and circles into a $2$-disk which allows to have cusps. A divide with cusps is the generalization of the notion of the divide which is introduced by A'Campo. From a divide with cusps, we can define the associated link in $S^3$. In this paper, we give the characterization of the link in $S^3$ which can be described as the associated link of a divide with cusps. In particular, we prove that every strongly invertible link and $2$-periodic link can be described as the link of a divide with cusps.

Divides with cusps and symmetric links

Abstract

A Divide with cusps is the image of a proper generic immersion from finite intervals and circles into a -disk which allows to have cusps. A divide with cusps is the generalization of the notion of the divide which is introduced by A'Campo. From a divide with cusps, we can define the associated link in . In this paper, we give the characterization of the link in which can be described as the associated link of a divide with cusps. In particular, we prove that every strongly invertible link and -periodic link can be described as the link of a divide with cusps.
Paper Structure (4 sections, 9 theorems, 11 equations, 17 figures)

This paper contains 4 sections, 9 theorems, 11 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Divides with cusps and associated links
  3. Involutions on links
  4. Main result and examples

Key Result

Theorem 1.1

A link $L$ in $S^3$ is described as the associated link of a divide with cusps if and only if there exists an involution $j$ on $S^3$ which has a non-empty fixed point set such that $j(L) = L$.

Figures (17)

  • Figure 1: An example of a divide with cusps
  • Figure 2: Smooth curve, upper cusp, lower cusp, and their links
  • Figure 3: Perturbation of a divide with cusps
  • Figure 4: An example of the diagram of $L(P)$
  • Figure 5: Local link diagrams of divides with cusps
  • ...and 12 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1
  • ...and 15 more