Band-unknotting numbers and connected sums of knots

Nakisa Ghanbarian; Stanislav Jabuka

paper

Band-unknotting numbers and connected sums of knots

arXiv:2512.06299

Abstract

We study the band-unknotting number $u_{nb} (K)$ of a knot $K$ , and how it behaves with respect to connect sums. We show that this sub-additive function is not additive under connected sums, by finding infinitely many examples of knots $K_{1}, K_{2}$ with $u_{nb} (K_{1} # K_{2}) < u_{nb} (K_{1}) + u_{nb} (K_{2})$ . Even more surprisingly, there are infinitely many examples of knots $K_{1}, K_{2}$ such that $u_{nb} (K_{1} # K_{2}) < u_{nb} (K_{i})$ , $i = 1, 2$ . Our work is motivated by the recent analogous results for the Gordian unknotting number by Brittenham and Hermiller \cite{BrittenhamHermiller}. We also prove new lower and upper bounds on the topological and smooth non-orientable 4-genus of a knot $K$ .