arXiv subjects

Clayton Shonkwiler

Publications and source records attributed to Clayton Shonkwiler.

At least 19 recordsLinked to original sources

arXiv2026

Hard unknots are often easy from a different perspective

Recent attempts to train AI models to recognize knots have produced millions of ``hard'' unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for the methods based on triangulations of the knot complement found in Regina and SnapPy, but more difficult for simplifiers based on diagrammatic moves. We present ReAPR (Re-embedding And Pass Rerouting), a semi-diagrammatic simplifier which alternates pass-move reduction on a knot diagram with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an n-crossing diagram, the minimum total variation is 2(n-k), where k is the least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (~2.6 million examples in total) in under 30 seconds of total CPU time, making it as effective as the best non-diagrammatic unknot recognition method (SnapPy) and about 60x faster. ReAPR is just as effective at simplifying diagrams of random knots, where it is regularly used to simplify diagrams with tens of millions of crossings.

math.GT↗

arXiv2026

Trefoil Probabilities and Polyhedra

We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure $dμ(x,y)=\frac{dx dy}{π (1+x^2+y^2)^2}$. Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $μ$-content of a random triangle. This yields a regular octahedral probability of $\frac{15}{4π^2}$ and a trefoil probability of $\frac{1}{16π^2}$.

math.GT↗

arXiv2026

Looping Animations Using the Modular Flow and Elliptic Functions

This paper describes an approach to generating looping animations using the modular flow and elliptic functions. The modular flow is a flow on lattices with many periodic orbits, and elliptic functions are meromorphic, doubly-periodic functions which can be visualized using domain coloring.

math.HO↗

arXiv2026

A Land of Oblique Duality for Frames and Probabilistic Frames

Functions or distributions used to sample and to reconstruct signals often occur in different domains, like the Dirac delta and a band-limited bump function in classical sampling. Oblique dual frames generalize this phenomenon. In this paper, we provide new tools to study oblique dual frames and introduce a probabilistic variant of oblique dual frames. We first present the oblique dual frame potential and show that it is minimized precisely when the oblique dual coincides with the canonical oblique dual. We then define oblique dual probabilistic frames and oblique approximately dual probabilistic frames. In particular, we prove that for a given oblique dual probabilistic frame, the associated oblique dual probabilistic frame potential is minimized if and only if the frame is tight and the oblique dual is canonical. Moreover, the tightness assumption can be removed when the minimization is restricted to oblique dual probabilistic frames of pushforward type. Finally, we investigate perturbations of oblique dual probabilistic frames and show that if a probability measure is sufficiently close to an oblique dual probabilistic frame pair in the $2$-Wasserstein topology, then it forms an oblique approximately dual probabilistic frame.

math.FA↗

arXiv2026

Random knotting in very long off-lattice self-avoiding polygons

We present experimental results on knotting in off-lattice self-avoiding polygons in the bead-chain model. Using Clisby's tree data structure and the scale-free pivot algorithm, for each $k$ between $10$ and $27$ we generated $2^{43-k}$ polygons of size $n=2^k$. Using a new knot diagram simplification and invariant-free knot classification code, we were able to determine the precise knot type of each polygon. The results show that the number of prime summands of knot type $K$ in a random $n$-gon is very well described by a Poisson distribution. We estimate the characteristic length of knotting as $656500 \pm 2500$. We use the count of summands for large $n$ to measure knotting rates and amplitude ratios of knot probabilities more accurately than previous experiments. Our calculations agree quite well with previous on-lattice computations, and support both knot localization and the knot entropy conjecture.

cond-mat.stat-mech↗

arXiv2025

Approximately Dual and Pseudo-Dual Probabilistic Frames

This paper studies properties of dual probabilistic frames -- in particular in relation to redundancy -- and introduces both approximately dual probabilistic frames and pseudo-dual probabilistic frames. We show that the canonical dual probabilistic frame is the only dual frame of pushforward type of a probabilistic frame with zero redundancy. Furthermore, we show that probabilistic frames with finite redundancy are atomic and finite. Approximately dual probabilistic frames generalize duality, with pseudo-duality being a further generalization. We introduce these concepts and prove certain structural results. In particular, every probabilistic frame has a discrete finite frame as an approximate dual.

math.FA↗

arXiv2025

Direct Sampling of Confined Polygons in Linear Time

We present an algorithm for sampling tightly confined random equilateral closed polygons in three-space which has runtime linear in the number of edges. Using symplectic geometry, sampling such polygons reduces to sampling a moment polytope, and in our confinement model this polytope turns out to be very natural from a combinatorial point of view. This connection to combinatorics yields both our fast sampling algorithm and explicit formulas for the expected distances of vertices to the origin. We use our algorithm to investigate the expected total curvature of confined polygons, leading to a very precise conjecture for the asymptotics of total curvature.

math.GT↗

arXiv2025

Factoring the Laplacian to understand topological polymers

A ring polymer is a random walk whose steps obey a single linear condition; their sum vanishes. Factoring the graph Laplacian into the product of the incidence matrix and its transpose allows us to show that for a more complicated network, the steps must lie in a linear subspace determined by the graph topology. This provides a useful new perspective on the James--Guth theory of phantom elastic networks. In particular, we formulate phantom networks which are free from the constraints of fixed crosslinks. For a given network the solution of the loop constraints makes the partition function finite-valued in the path integral formulation without applying any external forces or fixing any monomer positions. The resulting probability distribution on edge displacements is rotationally invariant, which is practically quite useful for generating unbiased random samples of edge displacements and monomer positions. Furthermore, one can exactly calculate many physical quantities such as correlation functions with respect to this distribution. Finally, this reformulation lends itself well to the case of non-Gaussian distributions. We illustrate this by computing the expected radius of gyration of a ring polymer in a wide variety of models.

cond-mat.stat-mech↗

arXiv2025

New Upper Bounds for Stick Numbers

We use a version of simulated annealing with knot-type preserving moves to find polygonal representatives of various knot types with low stick number. These give better bounds on stick numbers of prime knots through 10 crossings, and for the first time give a comprehensive table of stick number bounds on all knots through 13 crossings. These are equal to existing lower bounds (and hence determine the stick number exactly) for 19 knot types whose exact stick number was not known previously.

math.GT↗

arXiv2025

Sampling Finite Unit Norm Tight Frames Using Symplectic Geometry

Unit-norm tight frames in finite-dimensional Hilbert spaces (FUNTFs) are fundamental in signal processing, offering optimal robustness to noise and measurement loss. In this paper we introduce the Eigenlift algorithm for sampling random FUNTFs. Our approach exploits the symplectic geometry of the FUNTF space, which we characterize as a symplectic reduction of frame space by a symmetry group. We then define a Hamiltonian torus action on this reduced space whose momentum map induces a fiber bundle structure. The algorithm proceeds by sampling a point from the base space, which is a convex polytope, lifting it deterministically to a point on the corresponding fiber, then acting on this point by a random element of the torus to obtain a random FUNTF. We implement the method in Python and validate it in low-dimensional settings where it is computationally feasible to sample the base polytope via rejection sampling.

math.FA↗

arXiv2025

Optimization and the Topology of Spaces of Parseval Frames

A Parseval frame is a spanning set for a Hilbert space which satisfies the Parseval identity: a vector can be expressed as a linear combination of the frame whose coefficients are inner products with the frame vectors. There is considerable interest within the signal processing community in the structural properties of the space of finite-dimensional Parseval frames whose vectors all have the same norm, or which satisfy more general prescribed norm constraints. In this paper, we introduce a function on the space of arbitrary spanning sets that jointly measures the failure of a spanning set to satisfy both the Parseval identity and given norm constraints. We show that, despite its nonconvexity, this function has no spurious local minimizers, thereby extending the Benedetto--Fickus theorem to this non-compact setting. In particular, this shows that gradient descent converges to an equal norm Parseval frame when initialized within a dense open set in the associated matrix space. We then apply this result to study the topology of frame spaces. Using our Benedetto--Fickus-type result, we realize spaces of Parseval frames with prescribed norms as deformation retracts of simpler spaces, leading to explicit conditions which guarantee the vanishing of their homotopy groups. These conditions yield new path-connectedness results for spaces of real Parseval frames, generalizing the Frame Homotopy Theorem, which has seen significant interest in recent years.

math.FA↗

arXiv2025

An exact formula for the contraction factor of a subdivided Gaussian topological polymer

We consider the radius of gyration of a Gaussian topological polymer $G$ formed by subdividing a graph $G'$ of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of $G$ in terms of the number of subdivisions of each edge of $G'$ and a new weighted Kirchhoff index for $G'$. The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.

cond-mat.stat-mech↗

arXiv2024

On the average squared radius of gyration of a family of embeddings of subdivision graphs

Suppose we have an embedding of a graph $\mathbf{G}$ created by subdividing the edges of a simpler graph $\mathbf{G'}$. The edges of $\mathbf{G}$ can be divided into subsets which join pairs of ``junction'' vertices in $\mathbf{G'}$. The displacement vectors of the edges in each subset sum to the displacement between junctions. We can construct a family of embeddings of $\mathbf{G}$ with the same junction positions by rearranging the displacements in each group. In this paper, we show that the average (squared) radius of gyration of these embeddings is given by a simple formula involving a weighted (squared) radius of gyration of the positions of the junctions and the sum of the squares of the lengths of the edges of $\mathbf{G}$ and $\mathbf{G'}$. This ensemble of graph embeddings arises naturally in polymer science.

math.CO↗

arXiv2024

Geometric Approaches to Matrix Normalization and Graph Balancing

Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the space of square complex matrices whose global minimizers are normal matrices. We show that this functional, which we refer to as the non-normal energy, has incredibly well-behaved gradient descent dynamics: despite it being non-convex, we show that the only critical points of the non-normal energy are the normal matrices, and that its gradient descent trajectories fix matrix spectra and preserve the subset of real matrices. We also show that, even when restricted to the subset of unit Frobenius norm matrices, the gradient flow of the non-normal energy retains many of these useful properties. This is applied to prove that low-dimensional homotopy groups of spaces of unit norm normal matrices vanish; for example, we show that the space of $d \times d$ complex unit norm normal matrices is simply connected for all $d \geq 2$. Finally, we consider the related problem of balancing a weighted directed graph -- that is, readjusting its edge weights so that the weighted in-degree and out-degree is the same at each node. We adapt the non-normal energy to define another natural functional whose global minima are balanced graphs and show that gradient descent of this functional always converges to a balanced graph, while preserving graph spectra and realness of the weights. Our results were inspired by concepts from symplectic geometry and Geometric Invariant Theory, but we mostly avoid invoking this machinery and our proofs are generally self-contained.

math.DG↗

arXiv2024

A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space. This method improves on the moment polytope sampling algorithm of Cantarella, Duplantier, Shonkwiler, and Uehara (2016) and has (expected) time per sample quadratic in the number of edges in the polygon. We use our new sampling method and a new code for computing invariants based on the Alexander polynomial to investigate the probability of finding unknots among equilateral closed polygons.

cond-mat.stat-mech↗

arXiv2023

On the existence of Parseval frames for vector bundles

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles -- this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors -- and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of $G$-bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.

math.DG↗

arXiv2022

Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent

Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on understanding the global structure of collections of finite frames with prescribed properties, such as spaces of unit norm tight frames. We extend some of these results to the more general setting of fusion frames -- a fusion frame is a collection of subspaces of a finite-dimensional Hilbert space with the property that any vector can be recovered from its list of projections. The notion of tightness extends to fusion frames, and we consider the following basic question: is the collection of tight fusion frames with prescribed subspace dimensions path connected? We answer (a generalization of) this question in the affirmative, extending the analogous result for unit norm tight frames proved by Cahill, Mixon and Strawn. We also extend a result of Benedetto and Fickus, who defined a natural functional on the space of unit norm frames (the frame potential), showed that its global minimizers are tight, and showed that it has no spurious local minimizers, meaning that gradient descent can be used to construct unit-norm tight frames. We prove the analogous result for the fusion frame potential of Casazza and Fickus, implying that, when tight fusion frames exist for a given choice of dimensions, they can be constructed via gradient descent. Our proofs use techniques from symplectic geometry and Mumford's geometric invariant theory.

math.FA↗

arXiv2022

Toric Symplectic Geometry and Full Spark Frames

The collection of $d \times N$ complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces -- i.e., frames -- are used to give robust representations of complex-valued signals, so that geometrical and measure-theoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of $d$ of its columns gives a basis for $\mathbb{C}^d$? We show that the probability is one, generalizing recent work of Cahill, Mixon and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps -- certain spectral invariants of a frame -- and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature.

math.FA↗