Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

The Limit Point of the Pentagram Map and Infinitesimal Monodromy

The pentagram map takes a planar polygon $P$ to a polygon $P^{\prime }$ whose vertices are the intersection points of the consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons that converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper, we show that Glick’s operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick’s operator measures is the extent to which this perturbed polygon does not close up.

An ergodic algorithm for generating knots with a prescribed injectivity radius

The first provably ergodic algorithm for sampling the space of thick equilateral knots off-lattice, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It is an off-lattice generalization of the pivot algorithm. This move to an off-lattice model provides a huge improvement in power and efficacy in that samples can have arbitrary values for parameters such as the thickness constraint, bending angle, and torsion, while the lattice forces these parameters into a small number of specific values. This benefit requires working in a manifold rather than a finite or countable space, which forces the use of more novel methods in Markov–Chain theory. To prove the validity of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint which is used to show that the algorithm is ergodic. This ergodic sampling allows for a statistically valid method for estimating probability distributions of arbitrary functions on the space of thick knots.

An n-dimensional Klein bottle

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

Topological complexity of planar polygon spaces with small genetic code

We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of n-gons for all n, but primarily for those whose genetic codes, in the sense of Hausmann and Rodriguez, are moderately small.