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Dror Bar-Natan:
Talks:
# Balloons and Hoops and their Universal Finite Type Invariant, BF
Theory, and an Ultimate Alexander Invariant

## University of Oxford, January 21, 2013

**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 *π*_{2}, and hoops "act" on balloons as
*π*_{1} acts on *π*_{2}. We will
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 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.