Virtual Knot Theory

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DMS-0245588 Louis KauffmanThis project takes a broad approach to virtual knot theory. Knot theorystudies the embeddings of curves in three-dimensional space. Equivalentlyknot theory studies the embeddings of curves in athickened two dimensional sphere. Virtual knot theory studies theembeddings of curves in thickened surfaces of arbitrary genus,up to the addition and removal of empty handles from the surface. From thepoint of view of classical knot theory a virtual knotwill appear as the trajectory of a particle that sometimes abruptlydisappears from three dimensional space and reappears laterat another point in space. An example of such a trajectory would be asuperstring moving in three dimensional space, butoccasionally taking a detour into higher dimensions. Virtual knots have aspecial diagrammatic theory that makes handling themvery similar to the handling of classical knot diagrams. With thisapproach, one can generalize many structures in classical knottheory to the virtual domain, and use the virtual knots to test the limitsof classical problems such as the question whetherthe Jones polynomial detects knots and the classical Poincare conjecture.Counterexamples to these conjectures exist in thevirtual domain, and it is an open problem whether any of thesecounterexamples are equivalent (by addition and subtraction ofempty handles) to classical knots and links. Virtual knot theory is animportant domain to be investigated for its own sake and for a deeperunderstanding of classical knot theory.The principal investigator hopes that the above analog relationship with string theorywill bear fruit. Virtual braids are being used by the principalinvestigator to establish relationships among quantum computing, quantumentanglement and topological entanglement. It is a long-standing goal ofthis project to work with knots, physics andother natural sciences such as molecular biology.Generalizations such as the virtual knot theory havepotential for use in a wide variety of applications where there is acombination of topology, and combinatorially modeled physicality.In such applications, the topology is only part of the picture. One isdealing with systems that can be modeled in a discrete way sothat certain specified changes are allowed in the forms of the models. Thequestion that a topologist asks is: What is invariant under the changes?This question is significant in applications because it corresponds to thestable properties of molecular structures and to conserved quantities inthe physics. The approaches used in combinatorial topology can begeneralized for use in a wide variety of contexts. A good example of thisis seen in the use of knot theory in molecular biology where the allowedchanges are a combination of what the topologist regards as continuousdeformations coupled with discontinous changes corresponding to enzymaticaction and recombination. This has led to a vigorous interplay betweenknot theory and molecular biology. By asking these questions abouttopological relationship and the nature of knot theory, new insights andapplications in molecular biology, physics and quantum computing arecoming forth.