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What length of rope (of given diameter) is required to tie a particular knot? Or, to turn the problem around, given an embedded curve, how thick a regular neighborhood of the curve also is embedded?… (More)

The convex hull of a set K in space consists of points which are, in a certain sense, “surrounded” by K. WhenK is a closed curve, we define its higher hulls, consisting of points which are “multiply… (More)

There has been recent interest in knot energies among mathematicians and natural scientists. When discretized, such energies can lead to effective algorithms for recognizing when two curves represent… (More)

- Bohdan Senyuk, Qingkun Liu, +4 authors Ivan I. Smalyukh
- Nature
- 2013

Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids, a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being… (More)

- Jason Cantarella, Robert B. Kusner, John M. Sullivan
- Nature
- 1998

Applications of knots to the study of polymers have emphasized geometric measures on curves such as ‘energy’ and ‘rope length’, which, when minimized over different configurations of a knot, give… (More)

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two… (More)

- Robert B. Kusner, JOHN M. SULLIVANAbstract
- 1998

For each embedded constant mean curvature surface in R 3 with three ends and genus zero, we construct a conjugate cousin boundary contour in S 3. The moduli space of such contours is parametrized by… (More)

- Karsten Große-Brauckmann, Robert B. Kusner, John M. Sullivan
- Proceedings of the National Academy of Sciences…
- 2000

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… (More)

R. Schoen has asked whether the sphere and the cylinder are the only complete (almost) embedded constant mean curvature surfaces with finite absolute total curvature. We propose an infinite family of… (More)

We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and… (More)