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

Abstract

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.

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

β

A repackaging of β Sec. 9.4

β ₀

A reduction of M Sec. 9.3

δ

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

π

The MMA “of groups” Sec. 3.4

π

The fundamental invariant Sec. 2.3

π

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

τ

Tensorial interpretation map Sec. 8.1

ω

The wheels part of M/ ζ Sec.5

ω

The scalar part in β/β ₀ Sec. 9.3

Υ

Capping and sliding Sec.10.2

ζ

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

A

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

bch

Baker-Campbell-Hausdorff Sec. 4.2

$C_{u}^{\gamma }$

Conjugating a generator Sec. 4.2

CA

Circuit algebra Sec. 7.1

CW

Cyclic words Sec. 5.1

CW ^r

CW mod degree 1 Sec. 5.1

c

A “sink” vertex Sec. 9.1

c _u

A “c-stub” Sec. 9.1

div _u

The “divergence” F L→C W Sec. 5.1

dm$^{ab}_{c}$

Double/diagonal multiplication Sec. 3.2

FA

Free associative algebra Sec. 5.1

FL

Free Lie algebra Sec. 4.2

Fun(X→Y)

Functions X→Y Sec. 8.1

H

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

hm$^{xy}_{z}$

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

M

The “main” MMA Sec. 5.2

M ₀

The MMA of trees Sec. 4.2

MMA

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

m _u

Meridians Sec. 2.3

$m^{ab}_{c}$

Strand concatenation Sec 3.2

OC

Overcrossings commute

$\mathcal P^{\text {bh}}$

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

R

Ring of c-stubs Sec. 9.2

R ^r

R mod degree 1 Sec. 9.3

R1,R1’,R2,R3

Reidemeister moves Sec. 2.2, 7.1

$RC_{u}^{\gamma }$

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

rKBH

Ribbon knotted balloons&hoops Def. 2.1

S

Set of strand labels Sec. 2.2

T

Set of tail / balloon labels Sec. 2

t 𝜖 ^u

Units Ex. 2.2, Sec. 4.2,5.2

tha^ux

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

UC

Undercrossings commute Fig. 3

u-tangle

A usual tangle Sec. 2.2

$u\mathcal T$

All u-tangles Sec. 2.2

u, v, w

Tail / balloon labels Sec. 2

v-tangle

A virtual tangle Sec. 2.4

$v\mathcal T$

All v-tangles Sec. 2.4

w-tangle

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

x ⫽ f

Postfix evaluation Sec. 10.5

f ∖ x

Entry removal Sec. 10.5

x→a

Inline function definition Sec. 10.5

uv͆

“Top bracket form” Sec. 6

${\overset{\frown}{uv}}$

A cyclic word Sec. 6