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The unoriented band unknotting numbers of torus knots

Keisuke Himeno

TL;DR

The paper determines the unoriented band unknotting number $u^u_b(T_{p,q})$ for torus knots and proves it equals the pinch number $P(p,q)$. It achieves this by linking $u^u_b$ to the unoriented knot Floer torsion order ${ m Ord}'(T_{p,q})$ and showing ${ m Ord}'(T_{p,q})=P(p,q)$ through a combinatorial analysis of pinch moves on a $(1,1)$-diagram, exploiting the $L$-space knot structure of positive torus knots. The authors introduce a detailed framework using differential bigons in the universal cover to track changes in ${ m Ord}'$ under pinch moves and derive an inductive equality that confirms conjectured bounds. Consequently, $P(p,q)$, computable via a continued fraction expansion of $ rac{q}{p}$, exactly captures the unoriented band unknotting complexity of torus knots, bridging knot Floer theory with non-orientable surgery invariants.

Abstract

The unoriented band unknotting number of a knot is the minimum number of oriented or non-oriented band surgeries that turn the knot into the unknot. Batson introduced a certain non-oriented band surgery for a torus knot. The minimum number of these operations required to turn a torus knot into the unknot is called the pinch number, and it can be easily calculated from the parameters of the torus knot. In this paper, we show that the unoriented band unknotting number and the pinch number coincide for torus knots. In the proof, we use the torsion order of the unoriented knot Floer homology.

The unoriented band unknotting numbers of torus knots

TL;DR

The paper determines the unoriented band unknotting number for torus knots and proves it equals the pinch number . It achieves this by linking to the unoriented knot Floer torsion order and showing through a combinatorial analysis of pinch moves on a -diagram, exploiting the -space knot structure of positive torus knots. The authors introduce a detailed framework using differential bigons in the universal cover to track changes in under pinch moves and derive an inductive equality that confirms conjectured bounds. Consequently, , computable via a continued fraction expansion of , exactly captures the unoriented band unknotting complexity of torus knots, bridging knot Floer theory with non-orientable surgery invariants.

Abstract

The unoriented band unknotting number of a knot is the minimum number of oriented or non-oriented band surgeries that turn the knot into the unknot. Batson introduced a certain non-oriented band surgery for a torus knot. The minimum number of these operations required to turn a torus knot into the unknot is called the pinch number, and it can be easily calculated from the parameters of the torus knot. In this paper, we show that the unoriented band unknotting number and the pinch number coincide for torus knots. In the proof, we use the torsion order of the unoriented knot Floer homology.

Paper Structure

This paper contains 10 sections, 15 theorems, 13 equations, 24 figures.

Table of Contents

  1. Introduction
  2. The pinch move
  3. The unoriented knot Floer homology
  4. A $(1,1)$--diagram
  5. A $(1,1)$--diagram of a torus knot
  6. Regions
  7. The pinch move and the $(1,1)$--diagram
  8. Differential bigons in the universal cover of the $(1,1)$--diagram
  9. Points in differential bigons
  10. Proof of Theorem \ref{['thm_main']}

Key Result

Theorem 1.1

Let $p,q>1$ be relatively prime integers with $p$ odd. We assume $p<q$ if $q$ is odd. Suppose that such that Then, ${\rm u}^u_b(T_{p,q})=n$.

Figures (24)

  • Figure 1: The pinch move. Here the opposite edges of the square are identified.
  • Figure 2: (Left) A fundamental domain of the torus $\Sigma$ with the torus knot $T_{p,q}$ (in this case $T_{3,5}$) depicted, where the opposite edges are identified. The red and blue curves are $\alpha$ and $\beta$ respectively. The point $o$ at the initial position is represented by the black dot. Move it in the direction of the arrow. (Right) Stop the point $o$ just before touching $\alpha$. The arc $k_\alpha$ is represented by the dashed line.
  • Figure 3: (Left) Moving $o$ from $w$ toward $z$. (Right) The point $o$ reaches $z$. $k_\beta$ is represented by the solid line.
  • Figure 4: The curve $\beta$ in the left diagram moves according to the blue arrows, resulting in the right diagram that is a reduced $(1,1)$--diagram.
  • Figure 5: Orient $\alpha$ (with its copy) and the rainbow arcs.
  • ...and 19 more figures

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Bat14JV21
  • Theorem 2.2: Theorem 2.7 of JV21
  • Proposition 2.3
  • proof
  • Proposition 3.1: Corollary 1.9 of GM23
  • Remark 3.2
  • Lemma 4.1
  • proof
  • ...and 22 more