**Handout:**
twbh.html,
twbh.pdf,
twbh.png.
**Sources:** twbh.zip, pensieve.

There's also a **paper in progress**.

**Abstract.** Balloons are two-dimensional spheres. Hoops are one
dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave
much like the first and second fundamental groups of a topological space
- hoops can be composed like in *π _{1}*, balloons
like in

We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant ζ of KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.