# Knot Theory

Instructor: Dror Bar-Natan, drorbn@math.huji.ac.il, 02-658-4187.

Classes: Tuesdays 14:00-16:00 at Sprintzak 215.

Office hours: Sundays 14:00-15:00 in my office, Mathematics 309.

Agenda: To learn about finite type knot invariants, and especially about their multiply-proven but not-sufficiently-well understood fundamental theorem, whose different proofs relate to almost everything in mathematics.

More details: Finite type invariants are invariants of knots that can be regarded, in a natural sense, as polynomials on the space of all knots. It is a pretty powerful bunch of invariants, though they are better liked for their beauty rather than their power. By the fundamental theorem, they are classified by certain nice combinatorial objects called chord diagrams (close relatives of the Feynman diagrams of quantum field theory), modulo certain nice relations that are deeply related to Lie algebras. In the first part of the course we will talk about knots, knot invariants, finite type knot invariants, chord diagrams and their basic algebraic properties, and about the relationship with Lie algebras. We will then choose between either one of two of the approaches to proving the fundamental theorem, which, as we shall see, is equivalent to the construction of a universal finite type invariant:

The mystical/physical/geometrical approach:
One looks at the Feynman diagram expansion of a certain quantum field theory, and hoopla, we get a universal finite type invariant! Or else, one studies some strange systems of integrals over certain configuration spaces, or some odd "chopstick counting" problems, and hoopla again.

Key words and phrases: perturbation theory, Feynman diagrams, configuration spaces, compactification, differential forms and Stokes' theorem, degrees and general position, etc.

 Some key formulas: A key picture:

The algebraic/combinatorial approach:
One find algebraic contexts within which knot theory is finitely presented; namely, it is generated by some finitely many generators, modulo some finitely many relations. Then to construct a universal finite type invariant we only need to assign values to the generators so that the relations are satisfied. Different ways of doing this lead to different combinatorial/algebraic/topological pictures.

Key words and phrases: parenthesizations, associators, pentagons and hexagons, quasi-Hopf algebras, cohomology, PBW, triangulations and Pachner moves, planar algebras, etc.

 Some key formulas: Some key pictures:

Background image: A table of knots and links by Rob Scharein, taken from http://www.cs.ubc.ca/nest/imager/contributions/scharein/zoo/knotzoo.html.