Band-unknotting numbers and connected sums of knots

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 $u_{nb}(K_1\mathrm{\#}{K}_2)<u_{nb}(K_1)+u_{nb}(K_2)$. Even more surprisingly, there are infinitely many examples of knots $K_1,K_2$ such that $u_{nb}(K_1\mathrm{\#}{K}_2)<u_{nb}(K_i)$, $i=1,2$. 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 $K$.