  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 homotopy groups of a topological space - hoops can be composed as in
\pi_1, balloons as in \pi_2, and hoops "act" on balloons as \pi_1 acts on
\pi_2. We observe that ordinary knots and tangles in 3-space map into KBH in
4-space and become amalgams of both balloons and hoops.
  We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word)
-valued invariant \zeta of (ribbon) KBHs in terms of the said compositions and
action and we explain its relationship with finite type invariants. We
speculate that \zeta is a complete evaluation of the BF topological quantum
field theory in 4D. We show that a certain "reduction and repackaging" of \zeta
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 should be a
wonderful playground.
