相关论文: Geometry of Calugareanu's theorem

We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings,…

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the…

Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson…

Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat…

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed…

We study ribbons of vanishing Gaussian curvature, i.e., flat ribbons, constructed along a curve in $\mathbb{R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask…

Double Field Theory suggests to view the whole massless sector of closed strings as the gravitational unity. The fundamental symmetries therein, including the $\mathbf{O}(D,D)$ covariance, can determine unambiguously how the Standard Model…

We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredients. The first is a framework that allows us to think of such…

This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound,…

The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We…

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some…

It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the…

The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a…

Active feedback between geometry and physics is woven throughout the study of Nature at its fundamental level, and is of key importance in string theory. Methods of complex algebraic geometry in particular have brought about an unrivaled…

The twisting number of a ribbon knot $K$ is the minimal number of tangle replacements on the symmetry axis of $J \# -J$ for any knot $J$ that is required to produce a symmetric union diagram of $K$. We prove that the twisting number is…

This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-connection only we show…

The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2].…

We propose a new theory of gravitation, in which the affine connection is the only dynamical variable describing the gravitational field. We construct the simplest dynamical Lagrangian density that is entirely composed from the connection,…

\ \ We have recently developed methods for obtaining exact two-point resistance of the complete graph minus $N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with…

Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized…