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The bridge number of surface links and kei colorings

Kouki Sato, Kokoro Tanaka

Abstract

Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number $n$ for any integer $n \geq 4$. To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.

The bridge number of surface links and kei colorings

Abstract

Meier and Zupan introduced bridge trisections of surface links in as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number for any integer . To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.

Paper Structure

This paper contains 7 sections, 10 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kei colorings of tri-plane diagrams
  3. A review of tri-plane diagrams
  4. Kei colorings of tri-plane diagrams
  5. Relationship to original kei colorings
  6. Proof of Theorem \ref{['thm: lower bound']}
  7. Kei colorings of twist spun knots

Key Result

Theorem 1.1

For any integer $n \geq 4$, there exist infinitely many distinct surface knots with bridge number $n$.

Figures (2)

  • Figure 1: Surfaces associated to Reidemeister moves.
  • Figure 2: The coloring of each broken sheet is uniquely induced from its boundary.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm: main']}
  • Definition 2.1: MZ17
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 10 more