Knot Theory and Statistical Mechanics

Abstract

Certain algebraic relations used to solve models in statistical mechanics were key to describing a mathematical property of knots known as a polynomial invariant. This connection, tenuous at first, has since developed into a significant flow of ideas. The appearance of such common ground is not atypical of recent developments in mathematics and physics--ideas from different fields interact and produce unexpected results. Indeed, the discovery of the connection between knots and statistical mechanics passed through a theory intimately related to the mathematical structure of quantum physics. This theory, called von Neumann algebras, is distinguished by the idea of continuous dimensionality. Spaces typically have dimensions that are natural numbers, such as 2, 3 or 11, but in von Neumann algebras dimensions such as 2 or {pi} are equally possible. This possibility for continuous dimension played a key role in joining knot theory and statistical mechanics. In another direction, the knot invariants were soon found to occur in quantum field theory. Indeed, Edward Witten of the Institute for Advanced Study in Princeton, N.J., has shown that topological quantum field theory provides a natural way of expressing the new ideas about knots. This advance, in turn, has allowed a beautiful generalization about themore » invariants of knots in more complicated three-dimensional spaces known as three-manifolds, in which space itself may contain holes and loops.« less