Unknotting nonorientable surfaces

Abstract

Given a nonorientable, locally flatly embedded surface in the 4-sphere of nonorientable genus $h$, Massey showed that the normal Euler number lies in $\{-2h,-2h+4,\dots ,2h-4,2h\}$. We prove that every such surface with knot group of order two is topologically unknotted, provided that the normal Euler number is not one of the extremal values in Massey’s range. When $h$ is 1, 2, or 3, the same holds even with extremal normal Euler number; the $h=1$ case is due to Lawson. We also study nonorientable embedded surfaces in the 4-ball with boundary a knot $K$ in the 3-sphere, again where the surface complement has fundamental group of order two and nonorientable genus $h$. We prove that any two such surfaces with the same normal Euler number become topologically isotopic, rel. boundary, after adding a single tube to each. If the determinant of $K$ is trivial, we show that any two such surfaces are isotopic, rel. boundary, again provided that they have non-extremal normal Euler number, or that $h$ is 1, 2, or 3.