Unknotting Nonorientable Surfaces of Genus 4 and 5

Abstract

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert =1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$, then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus $6$ and $7$.