Unknotting number is not additive under connected sum

2025/06/30 by Mark Brittenham, Susan Hermiller, Brittenham, Mark +1 · 5 voices · 3 citations
Mathematics · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2506.24088

Abstract

We give the first examples of a pair of knots K₁,K₂ in the 3-sphere for which their unknotting numbers satisfy u(K₁#K₂)

Cited by

• Étude on the Delta-unknotting number
• Band-unknotting numbers and connected sums of knots
• A fast algorithm for the Hecke representation of the braid group, and applications to the computation of the HOMFLY-PT polynomial and the search for interesting braids

Discussions

• Quick! Everyone update their elementary knot theory syllabus! arxiv.org/abs/2506.24088 🧮 [bsky, 17 points, 1 comments]
• In math news, I just learned from Ian Agol that unknotting number has been shown to not be additive under connected sum. arxiv.org/abs/2506.24088 [bsky, 2 points, 0 comments]
• Furries like knots, right? arxiv.org/pdf/2506.24088 [bsky, 1 points, 1 comments]
• https://bsky.app/profile/m1ngus.bsky.social/post/3lxzagdytfk2m [bsky, 1 points, 0 comments]
• Wow! arxiv.org/pdf/2506.24088 [bsky, 0 points, 0 comments]

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• A remark on the counterexample to the unknotting number conjecture