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Abstract. I will be talking about the knotting of 1-dimensional objects (hoops) and 2-dimensional objects (balloons) in 4-dimensional space: about a nice set of operations that act on such knottings, and about a nice invariant that intertwines these operations and that is valued in some copies of the free Lie algebra in a number of generators (trees) plus the space of cyclic words (wheels).
Meta-Abstract. This will be a repeat of a talk that I've given before and that I plan to give again in Zurich on Thursday. But this time I'll take twice as long for it, and the idea will be to also use the language and the opportunity to converse with the people present about the various other topics I could talk about later in my 3-month visit to Geneva. These include: