To get a grade in this class, write a nifty
report on one of the topics below. Your report must be
mathematically non-empty (reading it, the other students in this class
should learn something new from it, beyond notation). Also, elegance
counts! You should aim for presentation quality - I hope to
eventually post your projects on this site (with your permissions, of
- Goussarov's alternative
definition of finite type invariants, as in his paper
Interdependent modifications of links and invariants of finite
degree, Topology 37-3 (1998).
- The relationship with the Chern-Simons quantum field theory, as for
example in my thesis and in many
later and better sources such as Dylan Thurston's thesis, arXiv:math.QA/9901110.
- The Le-Murakami
notion of q-tangles (arXiv:hep-th/9401016) or
the Bar-Natan notion of non associative tangles
(here; yikes, how much I regret this
name. "Paranthesized tangles" is much better).
- Write an elegant (and correct) exposition of how operations on
knotted objects and invariants of knotted objects induce operations on
chord diagrams (multiplying invariants, doubling a component, etc.).
This, at least partially, was done by Willerton.
- Something on Kontsevich's original proof of the existence of a
universal finite type invariant. See e.g. my paper On the Vassiliev Knot
Invariants, or the paper The Kontsevich
Integral by Chmutov and Duzhin.
- Something on Hutchings' approach to
the proof of the fundamental theorem, which is successful in the case
of braids but still open in the case of knots. See his paper Integration
of Singular Braid Invariants and Graph Cohomology or my review
of it (with Stoimenow) The Fundamental Theorem of
Vassiliev Invariants or my unfinished exposition Finite Type Invariants by the
- Anything else. You may find some ideas here or just drop by and talk to me.
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.