Physical Sciences and Mathematics Commons^™

The Expected Total Curvature Of Random Polygons, Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, Clayton Shonkwiler

Robert Kusner

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.

We then consider the symmetric measure on closed polygons of fixed total length constructed by …

On Thickness And Packing Density For Knots And Links, Robert Kusner

Robert Kusner

We describe some problems, observations, and conjectures concerning density of the hexagonal packing of unit disks in R2.thickness and packing density of knots and links in S3 and R3. We prove the thickness of a nontrivial knot or link in S3 is no more than 4 , the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in R3 or S3 is generally less than √12 , the density of the hexagonal packing of unit disks in R2.

Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends And Genus Zero, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan

Robert Kusner

We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

The Spinor Representation Of Surfaces In Space, Robert Kusner, Nick Schmitt

Robert Kusner

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan [32], which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K = T(M). Given a conformal immersion of M into R3, the unique spin strucure on S2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s1, s2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion …

Moduli Spaces Of Embedded Constant Mean Curvature Surfaces With Few Ends And Special Symmetry, Karsten Grosse-Brauckmann, Robert Kusner

Robert Kusner

We give necessary conditions on complete embedded cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are twodimensional varieties in the moduli spaces of general cmc surfaces. We characterize fundamental domains of our cmc surfaces by associated great circle polygons in the three-sphere.