Unknotting orientable surfaces

Abstract

It is shown that every locally flatly embedded genus $g \in \{1,2\}$ surface in the $4$-sphere with knot group $\mathbb{Z}$ is unknotted. The same proof establishes that any two genus $g \in \{1,2\}$ surfaces in $D^4$ with knot group $\mathbb{Z}$ and common boundary an Alexander polynomial one knot are isotopic rel. boundary. In previous work, the author and Powell reduced such unknotting problems to a question concerning the cancellation of $(-t)$-quadratic forms over $\mathbb{Z}[t^{\pm 1}]$, which was solved in genus $g \geq 3$ using work of Bass. In genus $g=2$, we observe that the same proof goes through using further work of Bass. In genus $g=1,$ the result instead follows from a statement in commutative algebra which was proved with the assistance of AI. Combined with earlier work of Freedman on locally flat spheres with knot group $\mathbb{Z}$, this shows that a locally flatly embedded orientable surface in $S^4$ is unknotted if and only if its knot group is $\mathbb{Z}$.