**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*.