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

TL;DR

This paper introduces a universal finite type invariant for knotted balloons and hoops in 4-space, connecting it to BF theory and an ultimate Alexander invariant with strong compositional and computational properties.

Contribution

It develops a new invariant for knotted balloons and hoops in 4-space, linking finite type invariants, BF theory, and a comprehensive Alexander invariant.

Findings

Invariant ta is related to finite type invariants.

Repackaged ta acts as an ultimate Alexander invariant.

Invariant ta has excellent composition and computational properties.

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 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.