Tunnel number one knots satisfy the Berge conjecture

Li, Tao; Moriah, Yoav; Pinsky, Tali

doi:10.2140/gt.2025.29.3271

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Tunnel number one knots satisfy the Berge conjecture

Tao Li, Yoav Moriah and Tali Pinsky

Geometry & Topology 29 (2025) 3271–3343

DOI: 10.2140/gt.2025.29.3271

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 nontrivial 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 and Lecuona for tunnel number one knots.

Keywords

Berge conjecture, doubly primitive, Heegaard splitting, lens space, genus-reducing surgery, Dehn surgery

Mathematical Subject Classification

Primary: 57M99, 57K10, 57K30

References

Publication

Received: 2 August 2023

Revised: 17 August 2024

Accepted: 31 October 2024

Published: 22 September 2025

Proposed: Cameron Gordon

Seconded: Mladen Bestvina, Ian Agol

Authors

Tao Li
Department of Mathematics Boston College Chestnut Hill, MA United States

Yoav Moriah
Department of Mathematics Technion Haifa Israel

Tali Pinsky
Department of Mathematics Technion Haifa Israel

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Volume 29, issue 6 (2025)