asOC:yet notUC:

=

ωεβ/antiq-ave

Alekseev

Torossian

vuu v

v v

a cb

newspeak!

u

u v

u

u v

u

u v

u v

y →,x

. . .

v uu v

− 227

y y

x →

− −

“the generators”

Kbh(T ;H).

“Ribbon-knottedballoonsand hoops”

u

x

v

y

S2

b

Riddle. People oftenstudy π1(X) = [S1, X ]and π2(X) = [S2, X ].Why not πT (X) :=[T,X ]?

S1

T

b

b

b

ωεβ:=http://www.math.toronto.edu/~drorbn/Talks/Zurich-130919

Dror Bar-Natan, Zurich, September 2013

Let T be a finite set of “tail labels” and H a finite set of“head labels”. Set

M1/2(T ;H) := FL(T )H ,

“H-labeled lists of elements of the degree-completed free Liealgebra generated by T”.

FL(T ) =

{

2t2 −1

2[t1, [t1, t2]] + . . .

}

/(

anti-symmetryJacobi

)

. . . with the obvious bracket.

Trees and Wheels and Balloons and Hoops

15 Minutes on Algebra

Operations M1/2 → M1/2.

Tail Multiply tmuvw is λ 7→ λ � (u, v → w), satisfies “meta-

associativity”, tmuvu � tmuw

u = tmvwv � tmuv

u .

Head Multiply hmxyz is λ 7→ (λ\{x, y}) ∪ (z → bch(λx, λy)),

where

bch(α, β) := log(eαeβ) = α+ β +[α,β]

2+

[α,[α,β]]+[[α,β],β]

12+ . . .

satisfies bch(bch(α, β), γ) = log(eαeβeγ) = bch(α,bch(β, γ))and hence meta-associativity, hmxy

x � hmxzx = hmyz

y � hmxyx .

Tail by Head Action thaux is λ 7→ λ � RCλxu , where

C−γu : FL → FL is the substitution u → e−γueγ , or more

precisely,

C−γu : u → e− ad γ(u) = u− [γ, u] +

1

2[γ, [γ, u]] − . . . ,

and RCγu = (C−γ

u )−1. Then Cbch(α,β)u = C

α�RC−βu

u �Cβu hence

RCbch(α,β)u = RCα

u � RCβ�RCα

uu hence “meta uxy = (ux)y”,

hmxyz � thauz = thaux � thauy � hmxy

z ,

and tmuvw �C

γ�tmuvw

w = Cγ�RC−γ

vu �Cγ

v � tmuvw and hence “meta

(uv)x = uxvx”, tmuvw � thawx = thaux � thavx � tmuv

w .

Wheels. Let M(T ;H) := M1/2(T ;H) × CW(T ), whereCW(T ) is the (completed graded) vector space of cyclic wordson T , or equaly well, on FL(T ):

vu vu

= −

Operations. On M(T ;H), define tmuvw and hmxy

z as before,and thaux by adding some J-spice:

(λ;ω) 7→ (λ, ω + Ju(λx)) � RCλx

u ,

where Ju(γ) :=

∫ 1

0ds divu(γ�RCsγ

u )�C−sγu , and

Theorem Blue. All blue identities still hold.

Merge Operation. (λ1;ω1)∗(λ2;ω2) := (λ1 ∪ λ2;ω1 + ω2).

Tangle concatenations → π1 ⋉ π2. With dmabc := thaab �

tmabc � hmab

c ,

dmabc

mabca b c

a b cdivu +

γ

λ =M1/2(u, v;x, y) =

Finite type invariants makesense in the usual way, and“algebra” is (the primitive part of) “gr” of “topology”.

T

H

balloons / tails

∞

Examples.

x y z

u v

S4

R4

hoops / heads

More on

satisfies R123, VR123, D, andno!

• δ injects u-knots into Kbh (likely u-tangles too).• δ maps v-tangles to Kbh; the kernel contains the above andconjecturally (Satoh), that’s all.• Allowing punctures and cuts, δ is onto.

δ

15 Minutes on Topology

IfX is a space, π1(X)is a group, π2(X)is an Abelian group,and π1 acts on π2.

K � hmxyz : K � thaux:

u v

x y

u v

z

K:

x y

w

Properties.• Associativities: mab

a � maca = mbc

b � maba , for m = tm, hm.

• “(uv)x = uxvx”: tmuvw � thawx = thaux � thavx � tmuv

w ,• “u(xy) = (ux)y”: hmxy

z � thauz = thaux � thauy � hmxyz .

K � tmuvw :

ConnectedSums.

∗Punctures & CutsOperations

newspeak!

“Meta-Group-Action”

ribbonembeddings

x

t

x

y

x

uu x

u

Shin Satoh

ǫx:

ρ−ux:ρ+ux:

ǫu:

δδ

δ

δ δ

“v”

ωεβ/metaWith Selmani,

See also ωεβ/tenn, ωεβ/bonn, ωεβ/swiss, ωεβ/portfolio

Repackaging. Given ((x → λux);ω), set cx :=∑

v cvλvx,replace λux → αux := cuλux

ecx−1cx

and ω → eω, use tu = ecu ,and write αux as a matrix. Get “β calculus”.

Why happy? An ultimate Alexander inva-riant: Manifestly polynomial (time and si-ze) extension of the (multivariable) Alexan-der polynomial to tangles. Every step of thecomputation is the computation of the inva-riant of some topological thing (no fishy Gaus-sian elimination). If there should be an Alexander invariant

with a computable algebraic categorification, it is this one!

See also ωεβ/regina, ωεβ/caen, ωεβ/newton.“God created the knots, all else intopology is the work of mortals.”

Leopold Kronecker (modified) www.katlas.org

ζ: ; 0

for trees

+ cyclic colourpermutations,

ζ is computable! ζ of the Borromean tangle, to degree 5:

Loose Conjecture. For γ ∈ K(T ;H),∫

DADBe∫B∧FA

∏

u

eOγu (B))⊗

x

holγx(A) = eτ (ζ(γ)).

That is, ζ is a complete evaluation of the BF TQFT.

Tensorial Interpretation. Let g be a finite dimensional Liealgebra (any!). Then there’s τ : FL(T ) → Fun(⊕T g → g)and τ : CW(T ) → Fun(⊕T g). Together, τ : M(T ;H) →Fun(⊕T g → ⊕Hg), and hence

eτ : M(T ;H) → Fun(⊕T g → U⊗H(g)).

ζ and BF Theory. (See Cattaneo-Rossi,arXiv:math-ph/0210037) Let A denote a g-connection on S4 with curvature FA, and B ag∗-valued 2-form on S4. For a hoop γx, let

holγx(A) ∈ U(g) be the holonomy of A along γx.

For a ball γu, let Oγu(B) ∈ g

∗ be (roughly) theintegral of B (transported via A to ∞) on γu.

Cattaneo

The Invariant ζ. Set ζ(ǫx) = (x → 0; 0), ζ(ǫu) = ((); 0), and

Theorem. ζ is (log of) the unique homomor-phic universal finite type invariant on Kbh.

(. . . and is the tip of an iceberg)

; 0

Paper in progress with Dancso, ωεβ/wko

The β quotient is M divi-ded by all relations that uni-versally hold when when g isthe 2D non-Abelian Lie alge-bra. Let R = QJ{cu}u∈T K andLβ := R ⊗ T with central R and with [u, v] = cuv − cvu foru, v ∈ T . Then FL → Lβ and CW → R. Under this,

µ → ((λx);ω) with λx =∑

u∈T

λuxux, λux, ω ∈ R,

bch(u, v) →cu + cv

ecu+cv − 1

(

ecu − 1

cuu+ ecu

ecv − 1

cvv

)

,

if γ =∑

γvv then with cγ :=∑

γvcv,

u�RCγu =

(

1 + cuγuecγ − 1

cγ

)−1

ecγu− cuecγ − 1

cγ

∑

v 6=u

γvv

,

divu γ = cuγu, and Ju(γ) = log(

1 + ecγ−1cγ

cuγu

)

, so ζ is

formula-computable to all orders! Can we simplify?

Trees and Wheels and Balloons and Hoops: Why I Care

May class: ωεβ/aarhus Class next year: ωεβ/1350Paper: ωεβ/kbh

Moral. To construct an M -valued invariant ζ of (v-)tangles,and nearly an invariant on Kbh, it is enough to declare ζ onthe generators, and verify the relations that δ satisfies.

β Calculus. Let β(T ;H) be

ω x y · · ·u αux αuy ·v αvx αvy ·... · · ·

∣

∣

∣

∣

∣

∣

∣

∣

∣

ω and the αux’s arerational functions invariables tu, one foreach u ∈ T .

,

tmuvw :

ω · · ·u αv β... γ

7→

ω · · ·w α+ β... γ

,

ω1 H1

T1 α1∗ω2 H2

T2 α2

=ω1ω2 H1 H2

T1 α1 0T2 0 α2

,

hmxyz :

ω x y · · ·... α β γ

7→ω z · · ·... α+ β + 〈α〉β γ

,

thaux :

ω x · · ·u α β... γ δ

7→

ωǫ x · · ·u α(1 + 〈γ〉/ǫ) β(1 + 〈γ〉/ǫ)... γ/ǫ δ − γβ/ǫ

,

where ǫ := 1+α, 〈α〉 :=∑

v αv, and 〈γ〉 :=∑

v 6=u γv, and let

R+ux :=

1 xu tu − 1

R−ux :=

1 xu t−1

u − 1.

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An "infrastructure project" is hard (and sometimes non-glorious) work that's done now and pays off later.

An example, and the most important one within knot theory, is the tabulation of knots up to 10 crossings. I think it precedes Rolfsen, yet the result is often called "the Rolfsen Table of Knots", as it is famously printed as an appendix to the famous book by Rolfsen. There is no doubt the production of the Rolfsen table was hard and non-glorious. Yet its impact was and is tremendous. Every new thought in knot theory is tested against the Rolfsen table, and it is hard to find a paper in knot theory that doesn't refer to the Rolfsen table in one way or another.

A second example is the Hoste-Thistlethwaite tabulation of knots with up to 17 crossings. Perhaps more fun to do as the real hard work was delegated to a machine, yet hard it certainly was: a proof is in the fact that nobody so far had tried to replicate their work, not even to a smaller crossing number. Yet again, it is hard to overestimate the value of that project: in many ways the Rolfsen table is "not yet generic", and many phenomena that appear to be rare when looking at the Rolfsen table become the rule when the view is expanded. Likewise, other phenomena only appear for the first time when looking at higher crossing numbers.

But as I like to say, knots are the wrong object to study in knot theory. Let me quote (with some variation) my own (with Dancso) "WKO" paper:

Studying knots on their own is the parallel of studying cakes and pastries as they come out of the bakery - we sure want to make them our own, but the theory of desserts is more about the ingredients and how they are put together than about the end products. In algebraic knot theory this reflects through the fact that knots are not finitely generated in any sense (hence they must be made of some more basic ingredients), and through the fact that there are very few operations defined on knots (connected sums and satellite operations being the main exceptions), and thus most interesting properties of knots are transcendental, or non -algebraic, when viewed from within the algebra of knots and operations on knots (see [ AKT-CFA]).

The right objects for study in knot theory are thus the ingredients that make up knots and that permit a richer algebraic structure. These are braids (which are already well -studied and tabulated) and even more so tangles and tangled graphs.

Thus in my mind the most important missing infrastructure project in knot theory is the tabulation of tangles to as high a crossing number as practical. This will enable a great amount of testing and experimentation for which the grounds are now still missing. The existence of such a tabulation will greatly impact the direction of knot theory, as many tangle theories and issues that are now ignored for the lack of scope, will suddenly become alive and relevant. The overall influence of such a tabulation, if done right, will be comparable to the influence of the Rolfsen table.

Aside. What are tangles? Are they embedded in a disk? A ball? Do they have an "up side" and a "down side"? Are the strands oriented? Do we mod out by some symmetries or figure out the action of some symmetries? Shouldn't we also calculate the affect of various tangle operations (strand doubling and deletion, juxtapositions, etc.)? Shouldn't we also enumerate virtual tangles? w-tangles? Tangled graphs?

In my mind it would be better to leave these questions to the tabulator. Anything is better than nothing, yet good tabulators would try to tabulate the more general things from which the more special ones can be sieved relatively easily, and would see that their programs already contain all that would be easy to implement within their frameworks. Counting legs is easy and can be left to the end user. Determining symmetries is better done along with the enumeration itself, and so it should.

An even better tabulation should come with a modern front-end - a set of programs for basic manipulations of tangles, and a web-based "tangle atlas" for an even easier access.

Overall this would be a major project, well worthy of your time.

K11n150

The interchange of I-95 and I-695,

northeast of Baltimore. (more)

From [AKT-CFA]

From [FastKh]

http://katlas.org/

(Source: http://katlas.math.toronto.edu/drorbn/AcademicPensieve/2012-01/)

The Most Important Missing Infrastructure Project in Knot TheoryJanuary-23-1210:12 AM

2012-01 Page 1