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Tunnel number one knots satisfy the Berge Conjecture

Tao Li, Yoav Moriah, Tali Pinsky

TL;DR

The paper proves that tunnel number one knots $K$ with irreducible exteriors in $M\in\{S^3,(S^2\times S^1)\#L(r,s)\}$ that admit a lens-space Dehn surgery must be doubly primitive, yielding the Berge Conjecture for $S^3$ and resolving Greene–Baker–Buck–Lecuona-type questions for $S^2\times S^1$. It develops a framework based on genus-two Heegaard splittings, introducing a $(\mathcal{P},\mathcal{D})$-pair and a complexity $c(P,D,\alpha,\gamma)$, then uses wave moves and train-tracks together with Whitehead graphs to constrain all possible configurations of a key curve $\delta$. Across four main geometric configurations of $\delta$ (three short-path configurations and a long-path one), the authors show that either a Berge-type Berge-Gabai stabilization occurs or the diagram forces a doubly primitive presentation, thereby proving the main theorem. Consequently, the Berge Conjecture is resolved for $S^3$ in the tunnel-number-one case, and related conjectures for $S^2\times S^1$-connected sums with lens spaces follow as corollaries. The results deepen our understanding of lens-space surgeries and the role of doubly primitive presentations in low-dimensional topology.

Abstract

Let $K$ be a tunnel number one knot in $M$ with irreducible knot exterior, where $M$ is either $S^3$, or a connected sum of $S^2\times S^1$ with any lens space. (In particular, this includes $M = S^2\times S^1$.) We prove that if a non-trivial Dehn surgery on $K$ yields a lens space, then $K$ is a doubly primitive knot in $M$. For $M = S^3$ this resolves the tunnel number one Berge Conjecture. For $M = S^2\times S^1$ this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots.

Tunnel number one knots satisfy the Berge Conjecture

TL;DR

The paper proves that tunnel number one knots with irreducible exteriors in that admit a lens-space Dehn surgery must be doubly primitive, yielding the Berge Conjecture for and resolving Greene–Baker–Buck–Lecuona-type questions for . It develops a framework based on genus-two Heegaard splittings, introducing a -pair and a complexity , then uses wave moves and train-tracks together with Whitehead graphs to constrain all possible configurations of a key curve . Across four main geometric configurations of (three short-path configurations and a long-path one), the authors show that either a Berge-type Berge-Gabai stabilization occurs or the diagram forces a doubly primitive presentation, thereby proving the main theorem. Consequently, the Berge Conjecture is resolved for in the tunnel-number-one case, and related conjectures for -connected sums with lens spaces follow as corollaries. The results deepen our understanding of lens-space surgeries and the role of doubly primitive presentations in low-dimensional topology.

Abstract

Let be a tunnel number one knot in with irreducible knot exterior, where is either , or a connected sum of with any lens space. (In particular, this includes .) We prove that if a non-trivial Dehn surgery on yields a lens space, then is a doubly primitive knot in . For this resolves the tunnel number one Berge Conjecture. For this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots.

Paper Structure

This paper contains 14 sections, 22 theorems, 20 equations, 38 figures.

Table of Contents

  1. Preliminaries
  2. Primitive and doubly primitive
  3. A notion of complexity for a $(\mathcal{P},\mathcal{D})$-pair
  4. Meridional disks and Whitehead graphs
  5. The planar surface and the curves $\alpha$, $\gamma$, $\delta$ and $\varepsilon$
  6. Fixing the decomposition of $\widehat{\Sigma}$
  7. Paths, Junctions, Traintracks and the curves $\delta, \varepsilon$
  8. Proof of the main theorem
  9. The curve $\delta$ takes three short paths
  10. $\delta$ takes four short paths
  11. The curve $\delta$ takes a long path
  12. $\delta$ takes one short path in each annulus, a special case
  13. $\delta$ takes one short path in each annulus, the general case
  14. The Proof

Key Result

Theorem 7

Let $K\subset M$ be a tunnel number one knot with irreducible knot exterior, where $M$ is either $S^3$ or $(S^2 \times S^1) \# L(r,s)$, (where $L(r,s)$ is any lens space). If a non-trivial Dehn surgery on $K$ yields a lens space, then $K$ is doubly primitive.

Figures (38)

  • Figure 1: The surfaces $\Delta_P$ and $\widehat{R}_d$ used to modify the Heegaard splitting.
  • Figure 2: The three graphs given in Och.
  • Figure 3: Dual Whitehead graphs: case (1)
  • Figure 4: Dual Whitehead graphs: case (2)
  • Figure 5: The decomposition of $\widehat{\Sigma}$ into two annuli and two rectangles
  • ...and 33 more figures

Theorems & Definitions (95)

  • Conjecture 2: The Berge Conjecture
  • Conjecture 3: The tunnel number one Berge Conjecture
  • Conjecture 4: The lens space Dehn surgery
  • Conjecture 6: Berge Conjecture for $S^2 \times S^1$
  • Theorem 7
  • Conjecture 8: Strong genus reducing surgery
  • Conjecture 9
  • Definition 1.1.1
  • Definition 1.1.2
  • Lemma 1.1.3
  • ...and 85 more