Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1210.6537

The Expected Total Curvature of Random Polygons

Jason Cantarella·Alexander Y Grosberg·Robert B. Kusner·Clayton Shonkwiler
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution. We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, we are able to prove that the expected value of total curvature for a closed n-gon is exactly π/2 n + (π/4) 2n/(2n-3). As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are unknotted.

Explore related subjects

math.DGcond-mat.stat-mechmath.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jason Cantarella, Alexander Y Grosberg, Robert B. Kusner, Clayton Shonkwiler. 2012-10-24. The Expected Total Curvature of Random Polygons. https://doi.org/10.1353/ajm.2015.0015

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗
arXiv2026

Isoparametric foliations and bounded geometry

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.

math.DG↗
arXiv2026

Minimal foliations, codimension-one stable norms, and a question of Bangert

We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension $n\ge3$, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On $\mathbb T^3$, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on $\mathbb T^3$ admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.

math.DG↗