2013/08/07 by Dror Bar-Natan, Bar-Natan, Dror
Computer Science · Mathematics · #57M25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Topological and Geometric Data Analysis #math.GT #math.QA #msc:57M25
paper · pdf · doi:10.48550/arxiv.1308.1721
53 pages, many pictures
openalex publication_date 2013/08/07 · arxiv created 2013/08/08 · arxiv updated 2013/08/09 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
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 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 ζof (ribbon) KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζis a complete evaluation of the BF topological quantum field theory in 4D. We show that a certain "reduction and repackaging" of ζis 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.