Acta Mathematica Vietnamica

, Volume 40, Issue 2, pp 271–329 | Cite as

Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant

Article

Abstract

Balloons are 2D spheres. Hoops are 1D 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 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. 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.

Keywords

2-knots Tangles Virtual knots w-tangles Ribbon knots Finite type invariants BF theory Alexander polynomial Meta-groups Meta-monoids 

Mathematics Subject Classification (2010)

57M25 

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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