Virtual Knot Theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

DMS-0245588 Louis Kauffman This project takes a broad approach to virtual knot theory. Knot theory studies the embeddings of curves in three-dimensional space. Equivalently knot theory studies the embeddings of curves in a thickened two dimensional sphere. Virtual knot theory studies the embeddings of curves in thickened surfaces of arbitrary genus, up to the addition and removal of empty handles from the surface. From the point of view of classical knot theory a virtual knot will appear as the trajectory of a particle that sometimes abruptly disappears from three dimensional space and reappears later at another point in space. An example of such a trajectory would be a superstring moving in three dimensional space, but occasionally taking a detour into higher dimensions. Virtual knots have a special diagrammatic theory that makes handling them very similar to the handling of classical knot diagrams. With this approach, one can generalize many structures in classical knot theory to the virtual domain, and use the virtual knots to test the limits of classical problems such as the question whether the Jones polynomial detects knots and the classical Poincare conjecture. Counterexamples to these conjectures exist in the virtual domain, and it is an open problem whether any of these counterexamples are equivalent (by addition and subtraction of empty handles) to classical knots and links. Virtual knot theory is an important domain to be investigated for its own sake and for a deeper understanding of classical knot theory. The principal investigator hopes that the above analog relationship with string theory will bear fruit. Virtual braids are being used by the principal investigator to establish relationships among quantum computing, quantum entanglement and topological entanglement. It is a long-standing goal of this project to work with knots, physics and other natural sciences such as molecular biology. Generalizations such as the virtual knot theory have potential for use in a wide variety of applications where there is a combination of topology, and combinatorially modeled physicality. In such applications, the topology is only part of the picture. One is dealing with systems that can be modeled in a discrete way so that certain specified changes are allowed in the forms of the models. The question that a topologist asks is: What is invariant under the changes? This question is significant in applications because it corresponds to the stable properties of molecular structures and to conserved quantities in the physics. The approaches used in combinatorial topology can be generalized for use in a wide variety of contexts. A good example of this is seen in the use of knot theory in molecular biology where the allowed changes are a combination of what the topologist regards as continuous deformations coupled with discontinous changes corresponding to enzymatic action and recombination. This has led to a vigorous interplay between knot theory and molecular biology. By asking these questions about topological relationship and the nature of knot theory, new insights and applications in molecular biology, physics and quantum computing are coming forth.