Unknotting number and connected sums: The knots $4_1$ and $5_1$

Mark Brittenham; Susan Hermiller

Unknotting number and connected sums: The knots $4_1$ and $5_1$

Geometric Topology 2026-01-27 v1

Authors: Mark Brittenham , Susan Hermiller

Abstract

We show that the knots $K\in\{4_1,5_1\}$ can be paired with a corresponding knot $K^\prime$ such that $u(K\#K^\prime)<u(K)+u(K^\prime)$ . As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot $K^\prime$ for the knot $3_1$ .

Keywords

unknotting number connected sum knot theory additivity knot invariants

Cite

@article{arxiv.2601.18757,
  title  = {Unknotting number and connected sums: The knots $4_1$ and $5_1$},
  author = {Mark Brittenham and Susan Hermiller},
  journal= {arXiv preprint arXiv:2601.18757},
  year   = {2026}
}

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