Related papers: Knotting Minimal Sets

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal…

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…

For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the…

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum…

Let $(Y,T)$ be a minimal suspension flow built over a dynamical system $(X,S)$ and with (strictly positive, continuous) ceiling function $f\colon X\to\R$. We show that the eigenvalues of $(Y,T)$ are contained in the range of a trace on the…

Given a compact topological dynamical system (X, f) with positive entropy and upper semi-continuous entropy map, and any closed invariant subset $Y \subset X$ with positive entropy, we show that there exists a continuous roof function such…

Given a compact surface $\mathcal{M}$ with a smooth area form $\omega$, we consider an open and dense subset of the set of smooth closed 1-forms on $\mathcal{M}$ with isolated zeros which admit at least one saddle loop homologous to zero…

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of…

The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we…

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…

The topological dynamics of the horocyclic flow $h_{\mathbb{R}}$ on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular, on such a surface, the flow $h_{\mathbb{R}}$ is minimal, or the minimal…

We show that in a prime, closed, oriented 3-manifold M, equivalent knots are isotopic if and only if the orientation preserving mapping class group is trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the more…

We give a short proof of Bing's characterization of $S^3$: a compact, connected 3-manifold $M$ is $S^3$ if and only if every knot in $M$ is isotopic into a ball.

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group.…

The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture…

For certain families of fluid flow, a new conserved quantity -- stream-helicity -- has been established.Using examples of linked and knotted streamtubes, it has been shown that stream-helicity does, in certain cases, entertain itself with a…

We describe an algorithm that recognizes some (perhaps all) intrinsically knotted (IK) graphs, and can help find knotless embeddings for graphs that are not IK. The algorithm, implemented as a Mathematica program, has already been used by…

Network embedding which encodes all vertices in a network as a set of numerical vectors in accordance with it's local and global structures, has drawn widespread attention. Network embedding not only learns significant features of a…

We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem…

We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable…