Loops on a punctured disk and knotted tubes in \(R^4\)
Seminar/Forum
Homotopy classes of loops on a twicepunctured disk (more generally a surface with boundary) admit a Lie bialgebra structure called the GoldmanTuraev Lie bialgebra. The Lie bracket and cobracket are defined in terms of intersections and selfintersections of curves. Combining results of Alekseev  Kawazumi  Kuno  Naef with results of the speaker and BarNatan, one obtains a surprising statement: "wellbehaved universal finite type invariants" of the (enhanced) GoldmanTuraev Lie bialgebra 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 KashiwaraVergne equations). This is clearly the wrong proof of a worthwhile theorem. But what is the right proof?
Presenter

Dr Zsuzsanna Dancso, University of Sydney