On Thickness and Packing Density for Knots and Links

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in \sp^3 and $\R^3$. We prove the thickness of a nontrivial knot or link in \sp^3 is no more than $\frac{\pi}{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in $\R^3$ or \sp^3 is generally less than $\frac{\pi}{\sqrt{12}}$, the density of the hexagonal packing of unit disks in $\R^2$.Comment: 6 pages; to appear in Contemporary Mathematics volume edited by Calvo, Millett & Rawdo

On the densest packing of polycylinders in any dimension

Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.Comment: Edited to reflect acknowledgements in the published versio

The Spinor Representation of Surfaces in Space

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan, 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 \bbR^3, the unique spin strucure on S^2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s_1,s_2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion of M under a suitable integrability condition, which for a minimal surface is simply that the spinor sections are meromorphic. A spin structure S also determines (and is determined by) the regular homotopy class of the immersion by way of a \bbZ_2-quadratic form q_S. We present an analytic expression for the Arf invariant of q_S, which decides whether or not the correponding immersion can be deformed to an embedding. The Arf invariant also turns out to be an obstruction, for example, to the existence of certain complete minimal immersions. The later parts of this paper use the spinor representation to investigate minimal surfaces with embedded planar ends. In general, we show for a spin structure S on a compact Riemann surface M with punctures at P that the space of all such (possibly periodic) minimal immersions of M\setminus P into \bbR^3 (upto homothety) is the the product of S^1\times H^3 with the Grassmanian of 2-planes in a complex vector space \calK of meromorphic sections of S. An important tool -- a skew-symmetric form \Omega defined by residues of a certain meromorphic quadratic differential on M -- lets us compute how \calK varies as M and P are varied. Then we apply this to determine the moduli spaces of planar-ended minimal spheres and real projective planes, and also to construct a new family of minimal tori and a minimal Klein bottle with 4 ends. These surfaces compactify in S^3 to yield surfaces critical for the \Moebius invariant squared mean curvature functional W. On the other hand, Robert Bryant has shown all W-critical spheres and real projective planes arise this way. Thus we find at the same time the moduli spaces of W-critical spheres and real projective planes via the spinor representation.Comment: latex, 37 pages plus appendice

Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends and Special Symmetry

We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional 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.Comment: latex2e, AMS-latex, 24 page

The Spinor Representation of Minimal Surfaces

The spinor representation is developed and used to investigate minimal surfaces in {\bfR}^3 with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield surfaces critical for the M\"obius invariant squared mean curvature functional $W$. On the other hand, all $W\!$-critical spheres and real projective planes arise this way. Thus we determine at the same time the moduli spaces of $W\!$-critical spheres and real projective planes via the spinor representation.Comment: 63 pages, dvi file only, earlier version is GANG preprint III.27 available via http://www.gang.umass.edu

A Gordian Pair of Links

We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.Comment: 2 pages, 1 figur