September 13: Knotted Trivalent Graphs, Tetrahedra and Associators. If knot theory was finitely presented, one could define knot invariants by assigning values to the generators so that the relations are satisfied. Well, some mild generalization of knot theory, the theory of knotted trivalent graphs, is finitely presented, as we will see in this talk. We will also see that the resulting theory is essentialy equivalent, though much more symmetrical and elegant and topological, to the Drinfel'd theory of associators. The talk will follow the handout at HUJI-001116. (joint with Dylan Thurston).
September 19: Knot Invariants, Associators and a Strange Breed of Planar Algebras. If knot theory was finitely presented, one could define knot invariants by assigning values to the generators so that the relations are satisfied. Well, knot theory is finitely presented, at least as a Vaughan Jones-style "planar algebra". We define a strange breed of planar algebras that can serve as the target space for an invariant defined along lines as above. Our objects appear to be simpler than the objects that appear in Drinfel'd theory of associators - our fundamental entity is the crossing rather than the re-association, our fundamental relation is the third Reidemeister move instead of the pentagon, and our "relations between relations" are simpler to digest than the Stasheff polyhedra. Yet our end product remains closely linked with Drinfel'd's theory of associators and possibly equivalent to it. The talk will follow the slides at Fields-010111; see also MSRI-001206. (joint with Dylan Thurston).
September 25, Goussarov Day: Bracelets and the Goussarov Filtration on the Space of Knots. Following Goussarov's paper "Interdependent Modifications of Links and Invariants of Finite Degree" (Topology 37-3 (1998)) I will describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory turns out to be equivalent to the standard one, it nevertheless has its own share of intrinsic beauty. See my paper Bracelets and the Goussarov Filtration of the Space of Knots.
September 26: Khovanov's Categorification of the Jones Polynomial. In two recent and very novel papers, arXiv:math.QA/9908171 and arXiv:math.QA/0103190, Khovanov finds a graded chain complex whose graded Euler characteristic is is the Jones polynomial, and proves that each individual homology group of this complex is a link invariant. His construction is very simple and elegant, and yet orthogonal to everything else we know about knot theory and hence extremely interesting. I plan to explain Khovanov's construction in about 2/3 of the time of the talk, and leave the rest for discussion. There will be a handout; see Calgary-010824 and also my paper On Khovanov's Categorification of the Jones Polynomial.