Loops on a punctured disk and knotted tubes in \(R^4\)
Homotopy classes of loops on a twice-punctured disk (more generally a surface with boundary) admit a Lie bi-algebra structure called the Goldman-Turaev Lie bi-algebra. The Lie bracket and co-bracket are defined in terms of intersections and self-intersections of curves. Combining results of Alekseev - Kawazumi - Kuno - Naef with results of the speaker and Bar-Natan, one obtains a surprising statement: "well-behaved universal finite type invariants" of the (enhanced) Goldman-Turaev Lie bi-algebra are in bijection with the same invariants of a very different topological structure: a class of knotted tubes in R^4. However, the bijection goes through representing both classes of invariants as solutions to certain equations in Lie theory (the Kashiwara-Vergne equations). This is clearly the wrong proof of a worthwhile theorem. But what is the right proof?
Dr Zsuzsanna Dancso, University of Sydney