University of Georgia
Mathematics Department Colloquium
Jason H Cantarella
University of Pennsylvania
February 22, 1999
The Writhing of Knots and the Helicity of Vector Fields
Abstract:
The writhing number of a knot measures the wrapping and
coiling of the knot around itself. The helicity of a vector field
measures the wrapping and coiling of orbits of the field around
each other. We ask three fundamental questions:
1) What is the maximum writhing number of a curve of length L,
surrounded by an embedded tubular neighborhood of radius R?
2) What is the maximum helicity of a vector field of energy E,
on a fixed domain in 3-space?
3) What is the maximum helicity of a vector field of energy E,
on any domain in 3-space of unit volume?
In this talk, we'll present an overview of results on each of these problems,
developing connections between writhing number and helicity, and between
the helicity of vector fields and their curl. On particular domains,
we'll be able to solve (2) completely, and draw pictures of the
helicity-maximizing fields. For problem (3), we'll derive restrictions on
the geometry of the "optimal" domain, and describe a series of numerical
experiments which lead us to a specific (conjectured) optimal domain.