Unknotting nonorientable surfaces

TL;DR

This paper investigates the topological properties of nonorientable surfaces embedded in 4-manifolds, establishing conditions under which such surfaces are unknotted or isotopic, based on their normal Euler number and fundamental group properties.

Contribution

It proves new unknottedness and isotopy results for nonorientable surfaces in 4-spheres and 4-balls, especially relating to their normal Euler number and fundamental group constraints.

Findings

Surfaces with knot group of order two are unknotted unless at extremal Euler numbers.

Any two surfaces with same non-extremal Euler number become isotopic after adding a tube.

For trivial determinant knots, all such surfaces are isotopic under certain conditions.

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 $\lbrace -2h,-2h+4,\ldots,2h-4,2h \rbrace$. 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 $3$, we prove the same holds even with extremal normal Euler number. 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 $3$.