α edition: KBH.pdf (last updated Fri, 17 May 2013 21:35:11 -0400) first edition: later
http://www.math.toronto.edu/~drorbn/papers/KBH / { chic1, chic2 }
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 as in π1, balloons as in π2, and hoops "act" on balloons as π1 acts on π2. 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 ζ of (ribbon) 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 should be a wonderful playground.
Related Mathematica Notebook. "The free-Lie meta-group-action structure" (Source, PDF).
Related Talks. Hamburg-1208, Regina-1206.
Related Scratch Work is under
Pensieve: KBH.