Dr. Louis Kauffman

Abstract:

In this talk we discuss how, starting with John Horton Conway’s skein theory of the Alexander polynomial, that new invariants of knots arose (the Jones polynomial among them) that are related to statistical mechanics. I will tell the story of how I discovered statistical mechanics summation models for the Alexander and Jones polynomials. The will continue, via the work of Ed Witten, to gauge theory and quantum field theory. 

This relationship of knot theory and physical theory is intimately tied with a mystery discovered by Herman Weyl in the early part of the 20th century. Weyl discovered that if one takes a line element A in spacetime as a differential 1-form, and writes down dA in the sense of the differential forms of Grassmann, then dA expresses the mathematical form of the Electromagnetic Field.

The field is expressed by the holonomy of the form A  around loops in spacetime. Weyl was so impressed with his observation that he suggested building a Geometry that would unify his line element A and the metric of General Relativity to make a unified field theory. But Einstein asked why should spacetime lengths change under transport? And the Weyl theory did not quite succeed. Yet it did succeed by the quantum reformulation of Fritz London, where the key was to see that the holonomy could represent a phase change in the quantum wave-function. Experimental confirmation of the influence of a gauge potential A on quantum interference came much later with the Aharonov-Bohm effect. Theoretical influence of this idea came with the generalization of A to a Lie algebra valued 1-form and the corresponding generalized gauge theories such as Yang-Mills theory. Then the physical field is not dA but dA + A^A and the holonomy remains important. Witten suggested the use of a spatial gauge A so that measuring its holonomy along a knot K would produce invariants such as the Jones polynomial. Witten understood that a formal answer required integration over all  the connections A. This integral of Witten is a functional integral in the quantum field theory associated with A. It has deep formal properties that inform indeed not only the Jones polynomial, but a host of other invariants as well, and the seeds of relationships with the three manifold invariants of Reshetikhin and Turaev, and the Vassiliev invariants of knots and links. Topological Quantum Field Theory was born. This is the story of a revolution in knot theory that started with Conway in 1969, focused by Jones and Kauffman, and began again with Witten in 1988. This talk will discuss these matters and will mention more recent developments such as Khovanov homology whose physical interpretations are not yet fully articulated. The talk will be self-contained and suitable for a general scientific audience. We will illustrate these key ideas in the relationship of knots and natural science with geometry, diagrams and dynamics.