Algebraic Structures on Spaces of Knots

University of California at Davis, August 13, 2001


I will discuss a number of topics related to algebraic constructions of universal finite type invariants. The idea is to find presentations of knot theory, or of some mild generalizations of knot theory, in terms of finitely many generators and relations, and then to construct a universal finite type invariant by setting its values on the generators so as the relations are satisfied. One such presentation involves knotted trivalent graphs, and is genuinely 3-dimensional. In this presentation the main generator turns out to be the standardly embedded tetrahedron and the relations are on one hand equivalent to the pentagon and hexagon relations of Drinfel'd's theory of associators and on the other hand they are closely related to the Biedenharn-Elliot identities of 6j-symbols and to the Pachner moves of the theory of triangulations. Another such presentation involves Vaughan Jones' notion of a planar algebra and leads to a crossing-centric constructions of a universal finite type invariant (as opposed to the now-standard associativity-centric construction). Much of what I will discuss is work in progress, and this my talk will point at several ``live ends'', unfinished problems that don't seem to be dead ends. (Joint with Dylan Thurston; see