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 π 1, balloons as in π 2, and hoops “act” on balloons as π 1 acts on π 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 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)




I wish to thank the people of the Knot at Lunch seminar in Toronto (Oleg Chterental, Karene Chu, Zsuzsanna Dancso, Peter Lee, Stephen Morgan, James Mracek, David Penneys, Peter Samuelson, and Sam Selmani) for hearing this several times and for making comments and suggestions. Further thanks to Zsuzsanna Dancso for pushing me to write this, to Martin Bridson for telling me about LOT groups and to Marcel Bökstedt, Eckhard Meinrenken, and Jim Stasheff for further comments. This work was partially supported by NSERC grant RGPIN 262178, by ITGP (Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics), an ESF RNP, and by the Newton Institute in Cambridge, UK.

Glossary of Notations (Greek letters, then Latin, then symbols)

α, β, γ

Free Lie series Sec. 4

α, β, γ, δ

Matrix parts Sec. 9.4


A repackaging of β Sec. 9.4


A reduction of M Sec. 9.3


A map \(u\mathcal T/v\mathcal T/w\mathcal T\to \mathcal K^{\text {rbh}}\) Sec. 2.2

δα, δβ, δγ

Infinitesimal free Lie series Sec. 10.4


Units Sec. 3.2


The MMA “of groups” Sec. 3.4


The fundamental invariant Sec. 2.3


The projection \(\mathcal K^{\text {rbh}}_{0}\to \mathcal K^{\text {rbh}}\) Prop. 3.6

\(\rho ^{\pm }_{ux}\)

±-Hopf links in 4D Ex. 2.2

\({\sigma ^{x}_{y}}\)

Re-labelling Sec. 10.5


Tensorial interpretation map Sec. 8.1


The wheels part of M/ ζ Sec.5


The scalar part in β/β 0 Sec. 9.3


Capping and sliding Sec.10.2


The main invariant Sec. 5


The tree-level invariant Sec. 4


A β-valued invariant Sec. 9.4


A β 0-valued invariant Sec. 9.3


The matrix part in β/β 0 Sec. 9.3

a, b, c

Strand labels Sec. 2.2

\(\text {ad}_{u}^{\gamma },\,\text {ad}_{u}\{\gamma \}\)

Derivations of FL Def. 105

\(\mathcal A^{\text {bh}}\)

Space of arrow diagrams Sec. 7.2


Baker-Campbell-Hausdorff Sec. 4.2

\(C_{u}^{\gamma }\)

Conjugating a generator Sec. 4.2


Circuit algebra Sec. 7.1


Cyclic words Sec. 5.1

CW r

CW mod degree 1 Sec. 5.1


A “sink” vertex Sec. 9.1


A “c-stub” Sec. 9.1


The “divergence” F LC W Sec. 5.1


Double/diagonal multiplication Sec. 3.2


Free associative algebra Sec. 5.1


Free Lie algebra Sec. 4.2


Functions XY Sec. 8.1


Set of head/hoop labels Sec. 2


Units Ex. 2.2, Sec. 4.2,5.2


Head delete Sec. 3,4.2,5.2


Head multiply Sec. 3,4.2,5.2

\(h{\sigma ^{x}_{y}}\)

Head re-label Sec. 3,4.2,5.2


The “spice” FL → CW Sec. 5.1

\(\mathcal K^{\text {rbh}}\)

All rKBHs Def. 2.1

\(\mathcal K^{\text {rbh}}_{0}\)

Conjectured version of \(\mathcal K^{\text {rbh}}\) Sec. 3.3


4D linking numbers Sec. 10.1


Longitudes Sec. 2.3


The “main” MMA Sec. 5.2


The MMA of trees Sec. 4.2


Meta-monoid-action Def. 3.2, Sec. 10.3.4


Meridians Sec. 2.3


Strand concatenation Sec 3.2


Overcrossings commute

\(\mathcal P^{\text {bh}}\)

Primitives of \(\mathcal A^{\text {bh}}\) Sec. 7.3


Ring of c-stubs Sec. 9.2


R mod degree 1 Sec. 9.3


Reidemeister moves Sec. 2.2, 7.1

\(RC_{u}^{\gamma }\)

Repeated \(C_{u}^{\gamma }\) / reverse \(C_{u}^{-\gamma }\) Sec. 4.2


Ribbon knotted balloons&hoops Def. 2.1


Set of strand labels Sec. 2.2


Set of tail / balloon labels Sec. 2


Units Ex. 2.2, Sec. 4.2,5.2


Tail by head action Sec. 3,4.2,5.2


Tail delete Sec. 3,4.2,5.2

\( {tm}^{uv}_{w}\)

Tail multiply Sec. 3,4.2,5.2

\(t{\sigma ^{x}_{y}}\)

Tail re-label Sec. 3,4.2,5.2

t, x, y, z

Coordinates Sec. 2


Undercrossings commute Fig. 3


A usual tangle Sec. 2.2

\(u\mathcal T\)

All u-tangles Sec. 2.2

u, v, w

Tail / balloon labels Sec. 2


A virtual tangle Sec. 2.4

\(v\mathcal T\)

All v-tangles Sec. 2.4


A virtual tangle mod OC Sec. 2.4

\(w\mathcal T\)

All w-tangles Sec. 2.4

x, y, z

Head / hoop labels Sec. 2


An \(\mathcal A^{\text {bh}}\)-valued expansion Sec. 7.4

Merge operation Sec. 3,4.2,5.2

Composition done right Sec. 10.5


Postfix evaluation Sec. 10.5


Entry removal Sec. 10.5


Inline function definition Sec. 10.5


“Top bracket form” Sec. 6


A cyclic word Sec. 6


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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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