7 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 walk…
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…
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…
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…
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…