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.
We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to...
Mikhail Karpukhin, R. Kusner, Peter McGrath et al.· 0 citations
Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid...
Let $M$ be a closed, oriented manifold. The group $\operatorname{Aut}_\bullet(M)$ of based self-equivalences of $M$ is a natural invariant in manifold classification, and Hambleton and Kreck's commutative braid gives a systematic way to compute it for $4$-manifolds. We extend their framework to $BO\langle n+1\rangle$-m...
Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi/p$. For every $2\leq p\leq8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Is...
We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Lutting...
In this paper we study the global geometry and classification of complete minimal surfaces with embedded ends and finite total curvature in the product space $\mathbb{H}^2 \times \mathbb{R}$. We describe the structure of embedded ends of such surfaces, constraining the combinatorics of their asymptotic polygons at infi...
