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On keen bridge splittings of links

Ayako Ido, Yeonhee Jang, Tsuyoshi Kobayashi

TL;DR

This work extends the notion of keenness and strong keenness from Heegaard splittings to bridge splittings of links, proving that strongly keen $(g,b)$-splittings of distance $n$ exist for all $g\ge0$, $b\ge1$, $n\ge1$ with the exceptions $(g,b)=(0,1)$ and $(g,b,n)=(0,3,1)$. The authors develop a framework based on curve complexes, arc-curve complexes, and subsurface projections to control distance realisation, and they construct explicit $(g,b)$-splittings using specialized $(3,1)$-manifold blocks to achieve prescribed distances while ensuring unique pairs realize the distance. They prove that certain low-genus cases behave differently (e.g., $(0,3)$-splittings with distance $1$ cannot be keen), and they provide a complete suite of results by case analysis for $n\ge2$ and $n=1$, including the critical $n=1$ instance where strong keenness holds in many settings. The paper also thoroughly analyzes distance-$0$ splittings and the $(0,3)$-distance-$1$ situation, linking to reducibility, stabilization, and connected-sum structures, and it integrates Farey-graph/Ladder constructions to produce infinite families of strongly keen $(0,2)$-splittings with arbitrary distance. Overall, the results yield a robust toolkit for constructing bridge splittings with controlled distance properties and highlight the finite-type implications for related Goeritz groups and mapping class interactions.

Abstract

In this paper, we extend the concept of {\it (strongly) keenness} for Heegaard splittings to bridge splittings, and show that, for any integers $g$, $b$ and $n$ with $g\ge 0$, $b\ge 1$, $n\ge 1$ except for $(g,b)=(0,1)$ and $(g,b,n)=(0,3,1)$, there exists a strongly keen $(g,b)$-splitting of a link with distance $n$. We also show that any $(0,3)$-splitting of a link with distance $1$ cannot be keen.

On keen bridge splittings of links

TL;DR

This work extends the notion of keenness and strong keenness from Heegaard splittings to bridge splittings of links, proving that strongly keen -splittings of distance exist for all , , with the exceptions and . The authors develop a framework based on curve complexes, arc-curve complexes, and subsurface projections to control distance realisation, and they construct explicit -splittings using specialized -manifold blocks to achieve prescribed distances while ensuring unique pairs realize the distance. They prove that certain low-genus cases behave differently (e.g., -splittings with distance cannot be keen), and they provide a complete suite of results by case analysis for and , including the critical instance where strong keenness holds in many settings. The paper also thoroughly analyzes distance- splittings and the -distance- situation, linking to reducibility, stabilization, and connected-sum structures, and it integrates Farey-graph/Ladder constructions to produce infinite families of strongly keen -splittings with arbitrary distance. Overall, the results yield a robust toolkit for constructing bridge splittings with controlled distance properties and highlight the finite-type implications for related Goeritz groups and mapping class interactions.

Abstract

In this paper, we extend the concept of {\it (strongly) keenness} for Heegaard splittings to bridge splittings, and show that, for any integers , and with , , except for and , there exists a strongly keen -splitting of a link with distance . We also show that any -splitting of a link with distance cannot be keen.
Paper Structure (26 sections, 20 theorems, 79 equations, 32 figures)

This paper contains 26 sections, 20 theorems, 79 equations, 32 figures.

Table of Contents

  1. Introduction and background materials
  2. Introduction
  3. Preliminaries
  4. Curve complexes
  5. Subsurface projections
  6. $(g,b)$-splittings
  7. Unique geodesics
  8. $(3,1)$-manifold pairs for the proof of Theorem \ref{['thm-1']}
  9. When $b\ge 2$
  10. When $g\ge 2$
  11. Proof of Theorem \ref{['thm-1']} when $n\ge 2$
  12. Proof of Theorem \ref{['thm-1']} when $n\ge 3$ and $b\ge 2$
  13. Proof of Theorem \ref{['thm-1']} when $n=2$ and $b\ge 2$
  14. Proof of Theorem \ref{['thm-1']} when $n\ge 2$, $g\ge 2$ and $b=1$
  15. Proof of Theorem \ref{['thm-1']} when $n\ge 2$, $g=1$ and $b=1$
  16. ...and 11 more sections

Key Result

Theorem 1.1

For any integers $g$, $b$ and $n$ with $g\ge 0$, $b\ge 1$, $n\ge 1$ except for $(g,b)=(0,1)$, $(0,2)$ and $(g,b,n)=(0,3,1)$, there exists a strongly keen $(g,b)$-splitting of a link with distance $n$.

Figures (32)

  • Figure 1: $(W_i, s_i)$ and $D_i$.
  • Figure 2: $\Delta$ and $\Delta'$.
  • Figure 3: $\beta"$.
  • Figure 4: $\alpha=\gamma_1\cup\gamma_2$, $\partial\Delta$ and $\partial\Delta'$.
  • Figure 5: $N(\alpha\cup \partial \Delta')$.
  • ...and 27 more figures

Theorems & Definitions (93)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2: cf. IJK1
  • Lemma 2.3
  • Remark 3.1
  • ...and 83 more