KNOTS, BRAIDS AND ELEMENTARY PARTICLES

Louis H. Kauffman
University of Illinois, Chicago
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In the 19-th century Lord Kelvin (Sir William Thompson) conceived a theory of atoms as knotted vortices in the luminiferous aether. Kelvin's energy for this theory led to the first tabulations of knots by the mathematicians Tait, Kirkman and Little and to the beginnings of the topological theory of knots and links in three dimensional space. In the intervening years the subject of topology in general and knot theory in particular underwent great growth. The Kelvin theory of vortex atoms was eventually abandoned, but the idea of deep relationships among knots and elementary particles has kept resurfacing in many different forms. For example, in the 1970's the physicist Herbert Jehle suggested that elementary particles were identifiable as quantized knotted electromagnetic flux. It has been recently argued the gluon flux may have knotted states. String theory, has toyed with the possiblity of knotted strings in some of its formulations, (but 1-dimensional membranes do not knot in high dimensions) but has also led to other connections with the mathematical theory of knots through the seminal work of Edward Witten. Loop quantum gravity, as formulated by Ashtekar, Smolin and Rovelli, began with a deep interrelationship with knots and the states of quantum gravity. The theory then went deeper and began to study the genesis of spacetime itself through spin-foams, generalizations of the Penrose spin networks. In another direction the Penrose spin networks can be generalized to include the topology of braiding and knotting to form a braided tensor category that supports structures in topological quantum field theory. We will talk about the consequences of the braided spin networks for constructing sufficient unitary transformations for quantum information theory and quantum mechanics, and for the construction of topological quantum computation. We will show how concepts of elementary particles are built into such theories, and we will discuss related research with Sundance Bilson-Thompson and Jonathan Hackett on framed braids and knotted surface models for elementary particles. From our point of view all these structures arise from the act of discrimination in the form of the making/emergence of a distinction. A distinction is a (logical) particle that interacts with itself to either produce itself or to vanish. This is the structure of awareness becoming aware of itself. We shall argue that the structure of thought that thinks itself is at the source of these models.