arXiv subjects

Clayton Shonkwiler

Publications and source records attributed to Clayton Shonkwiler.

At least 37 records · Page 2Linked to original sources

arXiv2022

Radius of Gyration, Contraction Factors, and Subdivisions of Topological Polymers

We consider the topologically constrained random walk model for topological polymers. In this model, the polymer forms an arbitrary graph whose edges are selected from an appropriate multivariate Gaussian which takes into account the constraints imposed by the graph type. We recover the result that the expected radius of gyration can be given exactly in terms of the Kirchhoff index of the graph. We then consider the expected radius of gyration of a topological polymer whose edges are subdivided into $n$ pieces. We prove that the contraction factor of a subdivided polymer approaches a limit as the number of subdivisions increases, and compute the limit exactly in terms of the degree-Kirchhoff index of the original graph. This limit corresponds to the thermodynamic limit in statistical mechanics and is fundamental in the physics of topological polymers. Furthermore, these asymptotic contraction factors are shown to fit well with molecular dynamics simulations.

cond-mat.stat-mech↗

arXiv2022

New Superbridge Index Calculations from Non-Minimal Realizations

Previous work used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. The present work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

math.GT↗

arXiv2022

Random graph embeddings with general edge potentials

In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models. A configuration of $n$ monomers in $\mathbb{R}^d$ can be written as a collection of $d$ coordinate vectors, each in $\mathbb{R}^n$. Our first main result is that entries from different coordinate vectors are uncorrelated, even when they are different coordinates of the same monomer. We predict that this property holds in realistic simulations and in actual polymer configurations (in the absence of an external field). Our second main contribution is a theorem explaining when and how a probability distribution on embeddings of a complicated graph may be pushed forward to a distribution on embeddings of a simpler graph to aid in computations. This construction is based on the idea of chain maps in homology theory. We use it to give a new formula for edge covariances in phantom network theory and to compute some expectations for a freely-jointed network.

cond-mat.stat-mech↗

arXiv2022

Admissibility and Frame Homotopy for Quaternionic Frames

We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exist, is it always possible to interpolate between any two while fixing their spectra and norms? In other words, the first question is the admissibility question for quaternionic frames and the second is a generalization of the frame homotopy conjecture. We give complete answers to both questions. For the first question, the existence criterion is exactly the same as in the real and complex cases. For the second, the non-empty spaces of quaternionic frames with specified frame spectrum and frame vector norms are always path-connected, just as in the complex case. Our strategy for proving these results is based on interpreting equivalence classes of frames with given frame spectrum as adjoint orbits, which is an approach that is also well-suited to the study of real and complex frames.

math.FA↗

arXiv2021

All Prime Knots Through 10 Crossings Have Superbridge Index $\leq 5$

This paper gives new upper bounds on the stick numbers of the knots $9_{18}$, $10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$, $10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\leq 5$, completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.

math.GT↗

arXiv2021

Three proofs of the Benedetto-Fickus theorem

In 2003, Benedetto and Fickus introduced a vivid intuition for an objective function called the frame potential, whose global minimizers are fundamental objects known today as unit norm tight frames. Their main result was that the frame potential exhibits no spurious local minimizers, suggesting local optimization as an approach to construct these objects. Local optimization has since become the workhorse of cutting-edge signal processing and machine learning, and accordingly, the community has identified a variety of techniques to study optimization landscapes. This chapter applies some of these techniques to obtain three modern proofs of the Benedetto-Fickus theorem.

math.MG↗

arXiv2020

Performance of the Uniform Closure Method for open knotting as a Bayes-type classifier

The discovery of knotting in proteins and other macromolecular chains has motivated researchers to more carefully consider how to identify and classify knots in open arcs. Most definitions classify knotting in open arcs by constructing an ensemble of closures and measuring the probability of different knot types among these closures. In this paper, we think of assigning knot types to open curves as a classification problem and compare the performance of the Bayes MAP classifier to the standard Uniform Closure Method. Surprisingly, we find that both methods are essentially equivalent as classifiers, having comparable accuracy and positive predictive value across a wide range of input arc lengths and knot types.

math.GT↗

arXiv2020

New Computations of the Superbridge Index

The knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge index for prime knots through 10 crossings.

math.GT↗

arXiv2020

Distributions of Distances and Volumes of Balls in Homogeneous Lens Spaces

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the $k$th moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.

math.DG↗

arXiv2020

Expected Distances on Manifolds of Partially Oriented Flags

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.

math.DG↗

arXiv2019

Knots with Exactly 10 Sticks

We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

math.GT↗

arXiv2019

New Stick Number Bounds from Random Sampling of Confined Polygons

The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.

math.GT↗

arXiv2019

Symplectic Geometry and Connectivity of Spaces of Frames

Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper we focus on the frame homotopy conjecture posed in 2002 and recently proved by Cahill, Mixon, and Strawn, which says that the space of finite unit norm tight frames is connected. We give a simple symplectic proof of a double generalization of the frame homotopy conjecture, showing that spaces of complex frames with arbitrary prescribed norms and frame operators are connected. To spark further investigation, we also suggest a number of fundamental questions in frame theory which seem amenable to a symplectic approach.

math.FA↗

arXiv2019

Stiefel manifolds and polygons

Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons -- or even to formalize what a random polygon is -- it is convenient to interpret each polygon as a point in some parameter space, essentially trading the complexity of the object for the complexity of the space. In this paper I describe such an interpretation where the parameter space is a Stiefel manifold and show how to exploit the geometry of the Stiefel manifold both to generate random polygons and to morph one polygon into another.

math.HO↗

arXiv2019

A fractal dimension for measures via persistent homology

We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(μ)$ for each homological dimension $i\ge 0$, assigned to a probability measure $μ$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $μ$ is supported on a compact subset of Euclidean space $\mathbb{R}^m$ for $m\ge2$, then Steele's work implies that $\mathrm{dim}_{\mathrm{PH}}^0(μ)=m$ if the absolutely continuous part of $μ$ has positive mass, and otherwise $\mathrm{dim}_{\mathrm{PH}}^0(μ)<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $μ$ in an i.i.d. fashion. To some measures $μ,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $μ$ is the uniform distribution over the unit interval, and conjecture that it exists when $μ$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

math.DS↗

arXiv2018

Open and closed random walks with fixed edgelengths in $\mathbb{R}^d$

In this paper, we consider fixed edgelength $n$-step random walks in $\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median, providing a natural map from open polygons to closed polygons of the same edgelength. Using this, we first prove that a natural reconfiguration distance to closure converges in distribution to a Nakagami$(\frac{d}{2},\frac{d}{d-1})$ random variable as $n \rightarrow \infty$. We then strengthen this to an explicit probabilistic bound on the distance to closure for a random $n$-gon in any dimension with any collection of fixed edgelengths $w_i$. Numerical evidence supports the conjecture that our closure map pushes forward the natural probability measure on open polygons to something very close to the natural probability measure on closed polygons; if this is so, we can draw some conclusions about the frequency of local knots in closed polygons of fixed edgelength.

cond-mat.stat-mech↗

arXiv2017

Spherical Geometry and the Least Symmetric Triangle

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.

math.MG↗

arXiv2017

Random Triangles and Polygons in the Plane

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.

math.MG↗