4,665 research outputs found
Curves of Finite Total Curvature
We consider the class of curves of finite total curvature, as introduced by
Milnor. This is a natural class for variational problems and geometric knot
theory, and since it includes both smooth and polygonal curves, its study shows
us connections between discrete and differential geometry. To explore these
ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.Comment: 25 pages, 4 figures; final version, to appear in "Discrete
Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 200
Cubic Polyhedra
A cubic polyhedron is a polyhedral surface whose edges are exactly all the
edges of the cubic lattice. Every such polyhedron is a discrete minimal
surface, and it appears that many (but not all) of them can be relaxed to
smooth minimal surfaces (under an appropriate smoothing flow, keeping their
symmetries). Here we give a complete classification of the cubic polyhedra.
Among these are five new infinite uniform polyhedra and an uncountable
collection of new infinite semi-regular polyhedra. We also consider the
somewhat larger class of all discrete minimal surfaces in the cubic lattice.Comment: 18 pages, many figure
On the Minimum Ropelength of Knots and Links
The ropelength of a knot is the quotient of its length and its thickness, the
radius of the largest embedded normal tube around the knot. We prove existence
and regularity for ropelength minimizers in any knot or link type; these are
curves, but need not be smoother. We improve the lower bound for the
ropelength of a nontrivial knot, and establish new ropelength bounds for small
knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and
corrections; new section on asymptotic growth of ropelength; several new
reference
The Second Hull of a Knotted Curve
The convex hull of a set K in space consists of points which are, in a
certain sense, "surrounded" by K. When K is a closed curve, we define its
higher hulls, consisting of points which are "multiply surrounded" by the
curve. Our main theorem shows that if a curve is knotted then it has a nonempty
second hull. This provides a new proof of the Fary/Milnor theorem that every
knotted curve has total curvature at least 4pi.Comment: 7 pages, 6 figures; final version (only minor changes) to appear in
Amer.J.Mat
Coplanar constant mean curvature surfaces
We consider constant mean curvature surfaces of finite topology, properly
embedded in three-space in the sense of Alexandrov. Such surfaces with three
ends and genus zero were constructed and completely classified by the authors
in arXiv:math.DG/0102183. Here we extend the arguments to the case of an
arbitrary number of ends, under the assumption that the asymptotic axes of the
ends lie in a common plane: we construct and classify the entire family of
these genus-zero coplanar constant mean curvature surfaces.Comment: 35 pages, 10 figures; minor revisions including one new figure; to
appear in Comm. Anal. Geo
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