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



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.


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

Mathematics Subject Classification (2010)



  1. 1.
    Alekseev, A., Torossian, C.: The Kashiwara-Vergne conjecture and Drinfel’d’s associators. Ann. Math. 175, 415–463 (2012). arXiv:0802.4300 MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bar-Natan, D: On the assiliev knot invariants. Topology 34, 423–472 (1995)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bar-Natan, D.: Finite type invariants of w-knotted objects IV: some computations, in preparationGoogle Scholar
  4. 4.
    Bar-Natan, D., Dancso, Z.: Finite type invariants of W-knotted objects I: braids, knots and the alexander polynomial. arXiv:1405.1956.
  5. 5.
    Bar-Natan, D., Dancso, Z.: Finite type invariants of W-knotted objects II: tangles and the kashiwara-vergne problem. arXiv:1405.1955.
  6. 6.
    Bar-Natan, D., Selmani, S.: Meta-monoids, meta-bicrossed products, and the Alexander polynomial. J. Knot Theory Ramifications 22–10 (2013). arXiv:1302.5689
  7. 7.
    Cattaneo, A.S., Rossi, C.A.: Wilson surfaces and higher dimensional knot invariants. Commun. Math. Phys. 256-3, 513–537 (2005). arXiv:math-ph/0210037 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duistermaat, J.J., Kolk, J.A. .: Lie groups. Springer (1999)Google Scholar
  9. 9.
    Habegger, N., Lin, X-S.: The classification of links up to link-homotopy. J. Am. Math. Soc. 3, 389–419 (1990)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Habegger, N., Masbaum, G.: The Kontsevich integral and Milnor’s invariants. Topology 39-6, 1253–1289 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Habiro, K., Kanenobu, T., Shima, A.: Finite type invariants of ribbon 2-knots. In: Nencka, H. (ed.) Low dimensional topology Cont. Math. vol. 233 pp. 187–196 (1999)Google Scholar
  12. 12.
    Habiro, K., Shima, A.: Finite type invariants of ribbon 2-knots, II. Topol Appl 111-3, 265–287 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Howie, J.: On the asphericity of ribbon disc complements. Trans. Am. Math. Soc. 289-1, 281–302 (1985)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Howie, J.: Minimal seifert manifolds for higher ribbon knots. Geom. Topol. Monogr. 1, 261–293. arXiv:math/9810185
  15. 15.
    Kanenobu, T., Shima, A.: Two filtrations of ribbon 2-knots. Topol. Appl. 121, 143–168 (2002)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kauffman, L.H.: Virtual knot theory. European J. Comb. 20, 663–690 (1999). arXiv:math.GT/9811028 MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kawauchi, A.: A survey of knot theory. Basel, Birkhäuser (1996)MATHGoogle Scholar
  18. 18.
    Kirk, P., Livingston, C., Wang, Z.: The gassner representation for string links. Comm. Cont. Math. 3, 87–136 (2001). arXiv:math/9806035 MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kricker, A.: The lines of the Kontsevich integral and Rozansky’s rationality conjecture. arXiv:math/0005284
  20. 20.
    Kuperberg, G.: What is a virtual link? Algebr. Geom. Topol. 3, 587–591 (2003). arXiv:math.GT/0208039 MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lee, P.: Proof of a conjectured formula for the Alexander invariant. arXiv:1209.0668
  22. 22.
    Meinrenken, E.: Lie groups and lie algebras. Lecture notes, University of Toronto (2010).
  23. 23.
    Milnor, J.W.: Link groups. Annals Math. 59, 177–195 (1954)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Reutenauer, C.: Free lie algebras. Clarendon, Oxford (1993)MATHGoogle Scholar
  25. 25.
    Rowen, L.H.: Polynomial identities in ring theory. Academic, New York (1980)MATHGoogle Scholar
  26. 26.
    Satoh, S.: Virtual knot presentations of ribbon torus knots. J. of Knot Theory and its Ramifications 9-4, 531–542 (2000)MathSciNetGoogle Scholar
  27. 27.
    Watanabe, T.: Clasper-moves among ribbon 2-knots characterizing their finite type invariants. J. of Knot Theory and its Ramifications 15-9, 1163–1199 (2006)CrossRefGoogle Scholar
  28. 28.
    Watanabe, T.: Configuration space integrals for long n-Knots, the Alexander polynomial and knot space cohomology. Alg. Geom. Top. 7, 47–92 (2007). arXiv:math/0609742 MATHCrossRefGoogle Scholar
  29. 29.
    Winter, B.: The classification of spun torus knots, vol. 18-9. arXiv:0711.1638 (2009)
  30. 30.
    Winter, B.: On codimension two ribbon embeddings. arXiv:0904.0684

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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