Unknotting number is not additive under connected sum

Mark Brittenham; Susan Hermiller

Unknotting number is not additive under connected sum

Geometric Topology 2025-09-16 v2

Authors: Mark Brittenham , Susan Hermiller

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, "Problems in low-dimensional topology", in the negative.

Keywords

knot theory

Cite

@article{arxiv.2506.24088,
  title  = {Unknotting number is not additive under connected sum},
  author = {Mark Brittenham and Susan Hermiller},
  journal= {arXiv preprint arXiv:2506.24088},
  year   = {2025}
}

Comments

Added new figures, and establish several new corollaries to the main theorem. Many small typographical changes

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