Band-unknotting numbers and connected sums of knots

TL;DR

This paper investigates the band-unknotting number's behavior under connected sums, revealing non-additivity and providing new bounds on the non-orientable 4-genus of knots, with implications for knot theory.

Contribution

It demonstrates that the band-unknotting number is not additive under connected sums and establishes new bounds on the non-orientable 4-genus of knots.

Findings

Counterexamples to additivity of band-unknotting number

Existence of knots with lower band-unknotting number after connected sum

New bounds on non-orientable 4-genus

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 $K$.