Knots and Polymers

``Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle.'' This is the text of Lewis Carroll’s Pillow Problem #58, from 1884. This and similar problems (e.g., ``what's the probability that a random quadrilateral is convex?'') sparked intense debate in the 19th century as mathematics was just starting to get to grips with the basics of geometric probability. Using the Grassmannian model of random polygons developed with Cantarella and Deguchi and based on work of Hausmann and Knutson, I will present the exact probability that a triangle is obtuse and that a polygon is convex, as well as precise statements about extreme and average triangles. Though this work is focused on planar polygons with few edges, it provides a template for integration and clustering in the physically-relevant setting of polygons in space. This is joint work with Laney Bowden, Jason Cantarella, Andrea Haynes, Tom Needham, Aaron Shukert, and Gavin Stewart. Statistical and Hydrodynamic Properties of Graph-shaped Polymers and Quaternions Tetsuo Deguchi Ochanomizu University We study statistical and hydrodynamic properties of graph-shaped polymers, which we call topological polymers, by applying the quaternion method for generating random polygons. We first review topological polymers such as double-ring polymers, tadpole polymers (lassos), and complete bipartite graph polymers, which are synthesized in chemistry. For simplicity, we regard ring polymers as topological polymers. We introduce the algorithm for generating random walks connecting given two points in the three-dimensional space, based on the Hopf map of quaternions [1, 2]. Surprisingly, the computational time of the algorithm is proportional to the number of steps in the random walks, that is, it gives a linear-time algorithm [1]. We compare the results of ideal topological polymers with those of real topological polymer obtained by the molecular dynamical simulation of the Kremer-Grest model. We suggest that for such graphs consisting of only up to trivalent vertices the statistical and hydrodynamic properties of ideal topological polymers are similar to those of the corresponding real topological polymers [2]. Finally, we also suggest enhancement of intra-chain two-point short-distance correlation for real topological polymers [2]. The talk is based on the collaboration with J. Cantarella, C. Shonkwiler and E. Uehara. [1] J. Cantarella, T. Deguchi, and C. Shonkwiler, Probability Theory of Random Polygons from the Quaternionic Viewpoint, Comm. Pure Appl. Math. Vol. 67, 1658-1699 (2014). [2] E. Uehara and T. Deguchi, Statistical and hydrodynamic properties of topological polymers for various graphs showing enhanced short-range correlation, J. Chem. Phys. Vol. 145, 164905 (2016). Topology for Polymeric Materials