Configuration spaces and imbedding problems. (English) Zbl 0874.57006

Bott, Raoul

Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. Proceedings of the conference at Aarhus University, Aarhus, Denmark, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 184, 135-140 (1997).

The author’s introduction: “The purpose of this talk is to present joint work with Clifford Taubes on a purely topological approach towards the recent physics-inspired self-linking invariants for knots described by Dror Bar-Natan and Guadagnini, Martinelli, and Mintchev. As I hope to show, the configuration spaces and their natural compactifications à la Fulton and MacPherson are precisely the needed ingredients to explain these invariants and their generalizations.”
Starting from the Gauss integral formula for the linking number of two circles in \(\mathbb{R}^3\), the author sketches a development leading to knot invariants which are also related to work of Kontsevich. The main underlying theme is the standard volume form \(\omega\) on \(S^2\) and integrating various products of pullbacks of \(\omega\) over appropriate compactifications of configuration spaces.
For the entire collection see [Zbl 0855.00020].

Reviewer: U.Koschorke (Siegen)

MSC:

57M25

Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

self-linking invariants; knots; configuration spaces; knot invariants