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A plat form presentation for surface-links

Jumpei Yasuda

Abstract

In this paper, we introduce a method, called a plat form, of describing a surface-link in the 4-space using a braided surface. We prove that every surface-link, which is not necessarily orientable, can be described in a plat form. The plat index is defined as a surface-link invariant, which is an analogy of the bridge index for a link in the 3-space. We classify surface-links with plat index $1$ and show some examples of surface-links in plat forms.

A plat form presentation for surface-links

Abstract

In this paper, we introduce a method, called a plat form, of describing a surface-link in the 4-space using a braided surface. We prove that every surface-link, which is not necessarily orientable, can be described in a plat form. The plat index is defined as a surface-link invariant, which is an analogy of the bridge index for a link in the 3-space. We classify surface-links with plat index and show some examples of surface-links in plat forms.

Paper Structure

This paper contains 10 sections, 18 theorems, 18 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A plat form presentation for a link
  4. Surface-links
  5. Braided surfaces and $2$-dimensional braids
  6. A plat form presentation for a surface-link
  7. Proofs of Theorems \ref{['Main theorem-A The existence of a plat form']} and \ref{['Main Theorem-B The existence of a genuine plat form']}
  8. Stabilization and generalized stabilization for braids
  9. A banded link presentation for a surface-link
  10. The plat index of a surface-link and examples

Key Result

Theorem \oldthetheorem

Every surface-link is equivalent to a surface-link in a plat form.

Figures (26)

  • Figure 1: A closed braid form.
  • Figure 2: A plat form.
  • Figure 3: The $2$-twist spun trefoil in a (normal) plat form.
  • Figure 4: The plat closure of a braid.
  • Figure 5: The isotopic deformation changing $\widetilde{\beta_f}$ to the plat closure of the trivial braid.
  • ...and 21 more figures

Theorems & Definitions (37)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem: Brendle-Hatcher2008
  • Proposition \oldthetheorem: Brendle-Hatcher2008
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Rudolph1983, Viro90
  • Lemma \oldthetheorem: cf. Kamada2002_book
  • Proposition \oldthetheorem: Kamada1994-01Viro90
  • Definition \oldthetheorem
  • ...and 27 more