28 citations · 270 across the 43 of their papers we have counts for
44 papers
Hard unknots are often easy from a different perspective
Jason Cantarella, Henrik Schumacher, Clayton Shonkwiler
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 wa…
Random knotting in very long off-lattice self-avoiding polygons
Jason Cantarella, Tetsuo Deguchi, Henrik Schumacher +2
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, fo…
New Upper Bounds for Stick Numbers
Jason Cantarella, Andrew Rechnitzer, Henrik Schumacher +1
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 s…
Optimization and the Topology of Spaces of Parseval Frames
Anthony Caine, Tom Needham, Clayton Shonkwiler
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…
Approximately Dual and Pseudo-Dual Probabilistic Frames
Dongwei Chen, Emily J. King, Clayton Shonkwiler
This paper studies properties of dual probabilistic frames -- in particular in relation to redundancy -- and introduces both approximately dual probabilistic frames and pseudo-dual…
An exact formula for the contraction factor of a subdivided Gaussian topological polymer
Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler +1
We consider the radius of gyration of a Gaussian topological polymer formed by subdividing a graph of arbitrary topology (for instance, branched or multicyclic). We give a…
On the average squared radius of gyration of a family of embeddings of subdivision graphs
Jason Cantarella, Henrik Schumacher, Clayton Shonkwiler
Suppose we have an embedding of a graph created by subdividing the edges of a simpler graph . The edges of can be divided into subsets which…
Geometric Approaches to Matrix Normalization and Graph Balancing
Tom Needham, Clayton Shonkwiler
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 s…
On the existence of Parseval frames for vector bundles
Samuel A. Ballas, Tom Needham, Clayton Shonkwiler
Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and importan…
A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting
Jason Cantarella, Henrik Schumacher, Clayton Shonkwiler
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 C…
Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent
Tom Needham, Clayton Shonkwiler
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 struc…
New Superbridge Index Calculations from Non-Minimal Realizations
Clayton Shonkwiler
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…
Random graph embeddings with general edge potentials
Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler +1
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.…
All Prime Knots Through 10 Crossings Have Superbridge Index
Clayton Shonkwiler
This paper gives new upper bounds on the stick numbers of the knots , , , , , , , , , , an…
Three proofs of the Benedetto-Fickus theorem
Dustin G. Mixon, Tom Needham, Clayton Shonkwiler +1
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…
Toric Symplectic Geometry and Full Spark Frames
Tom Needham, Clayton Shonkwiler
The collection of complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a fram…
Admissibility and Frame Homotopy for Quaternionic Frames
Tom Needham, Clayton Shonkwiler
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 t…
Performance of the Uniform Closure Method for open knotting as a Bayes-type classifier
Emily Tibor, Elizabeth M. Annoni, Erin Brine-Doyle +5
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 def…
New Computations of the Superbridge Index
Clayton Shonkwiler
The knots , , , , , , , , , , , , , , , , , , , and…
Distributions of Distances and Volumes of Balls in Homogeneous Lens Spaces
Brenden Balch, Chris Peterson, Clayton Shonkwiler
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-…
Radius of Gyration, Contraction Factors, and Subdivisions of Topological Polymers
Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler +1
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 appropria…
Expected Distances on Manifolds of Partially Oriented Flags
Brenden Balch, Chris Peterson, Clayton Shonkwiler
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…
Factoring the Laplacian to understand topological polymers
Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler +1
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 trans…
Knots with Exactly 10 Sticks
Ryan Blair, Thomas D. Eddy, Nathaniel Morrison +1
We prove that the knots and both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick num…