Knot Theory, Algebra, and Higher Algebra

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Why are mathematicians fascinated by the whole numbers? Certainly not because of the beauty inherent in staring at numbers such as 9,465,438. Neither is it due to the difficulty in figuring out that 9,465,438 is 2x3x1,577,573. The true reason is that the whole numbers are surprisingly deep, and the study of whole numbers, also known as "number theory", forces us to better understand, and indeed develop, many other useful and beautiful techniques, concepts and ideas. Number theory just seems to be related to everything.Likewise, though on a smaller scale, many knot theorists such as myself care little about shoelaces, yet care a lot about the unexpected ways by which the study of knotted shoelaces is intricately and deeply related to such a priori remote subjects as 3-dimensional manifolds, hyperbolic geometry, quantum field theory, differential geometry, Lie theory and representation theory, quantum algebra, combinatorics, homological algebra and sophisticated algorithmics.My research for this project will concentrate on the further elaboration of these unexpected links, using both analytical and computational tools. More specifically, my primary goal will be to complete our understanding of "homomorphic expansions" of classical and "virtual" knots. "Virtual knots" is a name for a certain type of knot theory in which the knots become much more of algebraic gadgets, rather than topological (yet certain classes of virtual knots describe certain classes of knots in 4-dimensional space). "Homomorphic expansions" are a certain class of knot invariants with deep connections to algebra and to quantum field theory, and the marriage of "virtual knots" to "homomorphic expansions" will likely benefit algebra by providing a unified framework for the study of all quantum groups.I tend to write expositions and give expository talks, draw pictures and write computer programs. Thus much of my work in this project will end up finding its way to my already-comprehensive web site, athttp://www.math.toronto.edu/~drorbn/.