KBH.pdf (last updated Wed, 04 Mar 2015 20:00:09 -0500)

arXiv:1308.1721
(updated less often)

to appear in Acta Mathematica Vietnamica

first edition: 07 Aug 2013

http://www.math.toronto.edu/~drorbn/papers/KBH /

{
bch,
chic1,
chic2,
ham,
mo,
ox,
tor,
viet
}

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 inπ, balloons as in_{1}π, and hoops "act" on balloons as_{2}πacts on_{1}π. We observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops._{2}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 Notebooks. **
"The free-Lie meta-monoid-action structure"
(Source,
PDF).
"A free-Lie calculator"
(Source,
PDF).

**Related Scratch Work** is under
Pensieve: KBH.