Michael Polyak
Department of Mathematics Technion - Israel Institute of Technology

Quadrisecants, homology intersections and finite type invariants.


Suppose that we managed to assign some signs to quadrisecants of a knot (i.e., lines cutting it 4 points) so that their total number, counted with signs, does not change under knot isotopy. What kind of invariant is it, and how to assign such signs? While the answer in this case is known, it motivates more general attempts to count (with signs) various geometric objects inscribed into a knot or a plane curve. In all cases invariants are easily seen to be of finite type. I will explain a general setting to produce such invariants (using evaluation maps of configuration spaces and homology intersections) and will formulate some results and conjectures. No prior knowledge of these themes will be assumed.