Unknotting number is not additive under connected sum

Mark Brittenham; Susan Hermiller

doi:https://doi.org/10.4007/annals.2026.204.2.7

Unknotting number is not additive under connected sum

Pages 795-817 from Volume 204 (2026), Issue 2 by Mark Brittenham, Susan Hermiller

Abstract

We give the first examples of knots $K_1$, $K_2$ in the $3$-sphere for which their unknotting numbers satisfy $u(K_1 \# K_2) \lt u(K_1)+ u(K_2)$. This answers question 1.69(B) from Kirby’s problem list, in the negative.

Keywords

connected sum, knot, unknotting number

Mathematical Subject Classification

Primary 2020: 57K10, 57K31

DOI

https://doi.org/10.4007/annals.2026.204.2.7

Milestones

Received: 15 October 2025
Revised: 16 February 2026
Accepted: 16 February 2025
Published online: 13 September 2026

Authors

Mark Brittenham

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE

Susan Hermiller

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE