Year: 2026
Type: preprint
Abstract: This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group $\mathbb{Z}_2$ and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in $D^4$ with knot group $\mathbb{Z}_2$ and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.
Source: arXiv (Cornell University)
Authors: Conway, Anthony, Kasprowski, Daniel
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206149974