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Clayton Shonkwiler

Publications and source records attributed to Clayton Shonkwiler.

48 records · Page 3Linked to original sources
arXiv2016

A Fast Direct Sampling Algorithm for Equilateral Closed Polygons

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.

cond-mat.stat-mech↗
arXiv2016

The symplectic geometry of closed equilateral random walks in 3-space

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname {SO}(3)$. The basic objects of study are the moment maps on equilateral random polygon space given by the lengths of any $(n-3)$-tuple of nonintersecting diagonals. The Atiyah-Guillemin-Sternberg theorem shows that the image of such a moment map is a convex polytope in $(n-3)$-dimensional space, while the Duistermaat-Heckman theorem shows that the pushforward measure on this polytope is Lebesgue measure on $\mathbb{R}^{n-3}$. Together, these theorems allow us to define a measure-preserving set of "action-angle" coordinates on the space of closed equilateral polygons. The new coordinate system allows us to make explicit computations of exact expectations for total curvature and for some chord lengths of closed (and confined) equilateral random walks, to give statistical criteria for sampling algorithms on the space of polygons and to prove that the probability that a randomly chosen equilateral hexagon is unknotted is at least $\frac{1}{2}$. We then use our methods to construct a new Markov chain sampling algorithm for equilateral closed polygons, with a simple modification to sample (rooted) confined equilateral closed polygons. We prove rigorously that our algorithm converges geometrically to the standard measure on the space of closed random walks, give a theory of error estimators for Markov chain Monte Carlo integration using our method and analyze the performance of our method. Our methods also apply to open random walks in certain types of confinement, and in general to walks with arbitrary (fixed) edgelengths as well as equilateral walks.

math.DG↗
arXiv2014

Homotopy Brunnian links and the $κ$-invariant

We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $μ$--invariants as homotopy periods of $\text{Conf}(n)$.

math.GT↗
arXiv2012

The Expected Total Curvature of Random Polygons

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.

math.DG↗
arXiv2012

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chordlengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chordlengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.

math.DG↗
arXiv2012

Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.

math.GT↗
arXiv2011

Pontryagin invariants and integral formulas for Milnor's triple linking number

To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus.

math.GT↗
arXiv2010

The complete Dirichlet-to-Neumann map for differential forms

The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition of the complete Dirichlet-to-Neumann map for differential forms in terms of two linear operators Φ and Ψ. The pair (Φ, Ψ) is equivalent to Joshi and Lionheart's operator Π and determines Belishev and Sharafutdinov's operator Λ. We show that the Betti numbers of the manifold are determined by Φ and that Ψ determines a chain complex whose homologies are explicitly related to the cohomology groups of the manifold.

math.DG↗
arXiv2009

Poincare duality angles for Riemannian manifolds with boundary

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and from Grassmannians, providing evidence that the Poincare duality angles measure, in some sense, how "close" a manifold is to being closed. This paper also elucidates a connection between the Poincare duality angles and the Dirichlet-to-Neumann operator for differential forms, which generalizes the classical Dirichlet-to-Neumann map arising in the problem of Electrical Impedance Tomography. Specifically, the Poincare duality angles are essentially the eigenvalues of a related operator, the Hilbert transform for differential forms. This connection is then exploited to partially resolve a question of Belishev and Sharafutdinov about whether the Dirichlet-to-Neumann map determines the cup product structure on a manifold with boundary.

math.DG↗
arXiv2009

Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r. To each such link L we associate a geometrically natural characteristic map g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g_L. Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer nu, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r. We show that the pairwise linking numbers p, q and r of the components of L are equal to the degrees of its characteristic map g_L restricted to the 2-dimensional subtori, and that twice Milnor's mu-invariant for L is equal to Pontryagin's nu-invariant for g_L. When p, q and r are all zero, the mu- and nu-invariants are ordinary integers. In this case we use J. H. C. Whitehead's integral formula for the Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with a formula for the fundamental solution of the scalar Laplacian on the 3-torus as a Fourier series in three variables, to provide an explicit integral formula for nu, and hence for mu.

math.GT↗
arXiv2008

Higher-dimensional linking integrals

We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.

math.GT↗