Unknotting number is not additive under connected sum
Abstract
We give the first examples of a pair of knots $K_1$, $K_2$ in the $3$-sphere for which their unknotting numbers satisfy $u(K_1 # K_2) < u(K_1)+u(K_2)$. This answers question 1.69(B) from Kirby's problem list, in the negative.
Fields of Study & Subjects
Geometric and Algebraic Topology
Geometric Analysis and Curvature Flows
Geometry and complex manifolds
Publication details
- Journal / source
- Annals of Mathematics
- Volume
- 204
- Issue
- 2
- Pages
- 795–817
- Publication date
- 1 September 2026
- Language
- EN