Related papers: Total curvature and isotopy of graphs in $R^3$

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve…

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the…

We define the total curvature of a semialgebraic embedding of a graph in the 3-dimensional Euclidean space. We prove that it satisfies a Chern-Lashof type inequality and we describe when the equality holds. We also prove a generalization of…

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of…

We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main…

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us…

In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For…

We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in 3-space, R the cross-section radius of a uniform tube neighborhood of K, L the arclength of K, and k the total…

A curve $\gamma$ in a Riemannian manifold $M$ is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $\gamma$ lies on an oriented…

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of…

Let $G$ be a simple topological graph and let $\Gamma$ be a polyline drawing of $G$. We say that $\Gamma$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of…

Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex whose facets are the complements of independent sets of size $k$ in $G$. We study the…

We prove that the total curvature of any planar graph with nonnegative combinatorial curvature is an integral multiple of $\frac{1}{12}.$ As a corollary, this answers a question proposed by T. R\'eti.

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $\Bbb H^2 \times \Bbb R$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$ which…

We provide a way to produce knots in $S^3$ from signed chord diagrams, and prove that every knot can be produced in this way. Using these diagrams, we generalize the fundamental theorem of finite type invariants. We also provide moves for…

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…

When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We…

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly…

An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet…