JCJason Cantarella

Jason Cantarella

University of Georgia
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Jason Cantarella has a total of 77 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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

    by Jason Cantarella, Tetsuo Deguchi, Henrik Schumacher, Clayton Shonkwiler, Erica Uehara

    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 $\num{656500} \pm \num{2500}$. 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.

    Cited by 2
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    New upper bounds for stick numbers

    by Jason Cantarella, Andrew Rechnitzer, Henrik Schumacher, Clayton Shonkwiler

    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.

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

    by Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, Erica Uehara

    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.

    Cited by 2
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    CoBarS: Fast Reweighted Sampling for Polygon Spaces in Any Dimension

    by Jason Cantarella, Henrik Schumacher

    We present the first algorithm for sampling random configurations of closed [Formula: see text]-gons with any fixed edgelengths [Formula: see text] in any dimension [Formula: see text] which is proved to sample correctly from standard probability measures on these spaces. We generate open [Formula: see text]-gons as weighted sets of edge vectors on the unit sphere and close them by taking a Möbius transformation of the sphere which moves the center of mass of the edges to the origin. Using previous results of the authors, such a Möbius transformation can be found in [Formula: see text] time. The resulting closed polygons are distributed according to a pushforward measure. The main contribution of the present paper is the explicit calculation of reweighting factors which transform this pushforward measure to any one of a family of standard measures on closed polygon space, including the symplectic volume for polygons in [Formula: see text]. For fixed dimension, these reweighting factors may be computed in [Formula: see text] time. Experimental results show that our algorithm is efficient and accurate in practice, and an open-source reference implementation is provided.

    Cited by 2
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    A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

    by 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 Cantarella et al (2016 J. Phys. A: Math. Theor. 49 275202) 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.

    Cited by 10
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    Radius of gyration, contraction factors, and subdivisions of topological polymers

    by Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, Erica Uehara

    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, which should be useful for predicting the g -factors of topological polymer models with excluded volume.

    Cited by 25
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    Families of similar simplices inscribed in most smoothly embedded spheres

    by Jason Cantarella, Elizabeth Denne, John McCleary

    Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$ . The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$ , where the factor in $\operatorname {O}(k)$ is a matrix called the pose. Among $(k-1)$ -spheres smoothly embedded in $\mathbb {R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$ . Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

    Cited by 3
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    Computing the Conformal Barycenter

    by Jason Cantarella, Henrik Schumacher

    The conformal barycenter of a point cloud on the sphere at infinity of the Poincaré ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edge lengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton--Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite-dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff--Ye iteration.

    Cited by 13
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    Families of similar simplices inscribed in most smoothly embedded spheres

    by Jason Cantarella, Elizabeth Denne, John McCleary

    Let $\Delta$ denote a non-degenerate $k$-simplex in $\mathbb{R}^k$. The set $\text{Sim}(\Delta)$ of simplices in $\mathbb{R}^k$ similar to $\Delta$ is diffeomorphic to $O(k)\times [0,\infty)\times \mathbb{R}^k$, where the factor in $O(k)$ is a matrix called the {\em pose}. Among $(k-1)$-spheres smoothly embedded in $\mathbb{R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\text{Sim}(\Delta)$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in O(k)$. Further, the intersection of $\text{Sim}(\Delta)$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $O(k)$ via the pose map. This gives a high dimensional generalization of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

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    Configuration Spaces, Multijet Transversality, and the Square-Peg Problem

    by Jason Cantarella, Elizabeth Denne, John McCleary

    We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold $M$ in Euclidean space, we can find a dense set of smooth embeddings of $M$ for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in $\mathbb{R}^n$ where each simple closed curve has an odd number of inscribed square-like quadrilaterals.

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