V5A1 - Advanced Topics in Algebra - Vassiliev invariants and related topics

Course description


In 1990, Vassiliev introduced a beautiful class of knot invariants that arise via Alexander duality from a spectral sequence related to a filtration on the space of singular knots. Remarkably, the essence of (the degree 0 part of) his construction can be captured in rather elementary terms: the idea is to extend a given knot invariant to singular knots by taking iterated differences of the various resolutions of a singular knot. Thinking of these iterated differences as discrete versions of derivatives, Vassiliev proposes to study those knot invariants that are "polynomial" in the sense that, for some natural number n, their nth derivatives vanish. The resulting theory of what are now called Vassiliev invariants has striking relations to various other subjects. In this course, we will focus on the following topics:


The textbook

and the following original references make good starting points. More references will be added to this list as the course progresses.



Familiarity with algebraic topology and homological algebra.

Final Exam

Oral exam