Related papers: Bridge spheres for the unknot are topologically mi…

We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in $S^3$, and in particular those that are unknotted or slice in $S^3$. We completely characterize all such curves for most twist knots: they…

In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be…

In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an…

We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. We then apply the procedure…

We prove a relatively simple combinatorial characterization of simplicial $d$-spheres on $d+4$ vertices. Our criteria are given in terms of the intersection patterns of a simplicial complex's family of minimal non-faces. Namely, let…

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of…

Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine the…

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of…

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…

We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to…

In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal…

we construct a properly embedded minimal surface in the flat product R^2*S^1 which is quasi-periodic but is not periodic.

Let TC$_n$(X) denote the n-th topological complexity of a topological space X. It is known that TC$_n$(X) does not exceed n-1 for non-contractible X, and so it makes sense to describe spaces X with TC$_n$(X) =n-1. Grant--Lupton--Oprea…

We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres…

Internal stabilization adds a trivial handle to an embedded surface in a coordinate chart. It is known that any pair of smoothly knotted surfaces in a simply-connected $4$-manifold become smoothly isotopic after sufficiently many internal…

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and…

We prove that each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.

We characterize the homotopically trivial non-contractible topological spaces with the minimum number of points.

In this paper, we give the trivializing number of all minimal diagrams of positive 2-bridge knots, and study the relation between the trivializing number and the unknotting number for a part of these knots.

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…