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 $Z_2$.

In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary K such that $|\det (K\left)\right|=1$, 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 $Z_2$ of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.