2024/02/28 by Pencovitch, Mark
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2402.18290
Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in D4 with knot group ℤ2. In particular we show that if two such surfaces have fixed knot boundary K in S4 such that \vert det(K) \vert =1, the same normal Euler number, 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 ℤ2 of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.