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

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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.

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  1. The bulk of the paper easily generalizes to the case where H (not T!) is infinite, though nothing is gained by allowing H to be infinite.

  2. The adjective “ribbon” will be explained in Definition 2.4.

  3. See “notational conventions”, Section 10.5.

  4. Better English would be “ordinary tangle”, but I want the short form to be “u-tangle”, which fits better with the “v-tangles” and “w-tangles” that arise later in this paper.

  5. In [16], the mixed/detour move was yet unnamed, and was simply “move (c) of Fig. 2”.

  6. See “notational conventions”, Section 10.5.

  7. We feel that the clarity of this paper is enhanced by this omission.

  8. \(tm^{uv}_{w}\), for example, is defined on M(T;H) exactly when u, vT yet wT ∖ {u, v}. All other operations behave similarly.

  9. I ignore set-theoretic difficulties. If you insist, you may restrict to countable groups or to finitely presented groups.

  10. A similar statement can be made for Alexander formulae based on the Burau representation. Yet note that such formulae still end with a computation of a determinant which may take O(n 3) steps. Note also that the presentation of knots as braid closures is typically inefficient—typically a braid with O(n 2) crossings is necessary in order to present a knot with just n crossings.

  11. Here and below, “drawn on Σ” means “embedded in Σ × [−𝜖, 𝜖]”.

  12. Following a private discussion with Dylan Thurston.

  13. A monoid is a group sans inverses. You lose nothing if you think “group” whenever the discussion below states “monoid”.

  14. Or merely an algebra.

  15. Think “group-action”.

  16. There are also ∗, t η u, h η x, \(t{\sigma ^{u}_{v}}\) and \(h{\sigma ^{x}_{y}}\) and units t 𝜖 u and h 𝜖 x as before.


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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.

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Correspondence to Dror Bar-Natan.

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Web resources for this paper are available at [Web/]:=, including an electronic version, source files, computer programs, lecture handouts and lecture videos. Throughout this paper, we follow the notational conventions and notations outlined in Section 10.5.

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

β 0

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

δ α, δ β, δ γ

Infinitesimal free Lie series Sec. 10.4

𝜖 a

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

ζ 0

The tree-level invariant Sec. 4

ζ β

A β-valued invariant Sec. 9.4

ζ β0

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

c u

A “c-stub” Sec. 9.1

div u

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

h 𝜖 x

Units Ex. 2.2, Sec. 4.2,5.2

h η

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

J u

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

l u x

4D linking numbers Sec. 10.1

l x

Longitudes Sec. 2.3


The “main” MMA Sec. 5.2

M 0

The MMA of trees Sec. 4.2


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

m u

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 r

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

t 𝜖 u

Units Ex. 2.2, Sec. 4.2,5.2


Tail by head action Sec. 3,4.2,5.2

t η u

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

Z bh

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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Bar-Natan, D. Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant. Acta Math Vietnam 40, 271–329 (2015).

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