Some Very Good Formulas for the Alexander Polynomial, 1 Abstract

Dror Bar-Natan: Talks: Oberwolfach-1405: ωεβ≔http://www.math.toronto.edu/˜drorbn/Talks/Oberwolfach-1405/ Some very good formulas for the Alexander polynomial, 1 Abstract. I will describe some very good formulas for a (matrix plus Meta-Associativity Runs. scalar)-valued extension of the Alexander polynnomial to tangles, then say that everything extends to virtual tangles, then roughly to simply knotted balloons and hoops in 4D, then the target space extends to (free Lie algebras plus cyclic words), and theresult is a universalfinite type of the knotted objects in its domain. Taking a cue from the BF topological R3 ... divide and conquer! quantum field theory, everything should extend (with some modifica- 2 tions) to arbitrary codimension-2 knots in arbitrary dimension and in 6 5 particular, to arbitrary 2-knots in 4D. But what is really going on is still 4 1 3 a mystery. 2 1 Tangles. 4 6 ⊔ r 5 TS, S T 3

a b c b ab g mc T T +6 ++10 + 13 14 15 16 Why Tangles? 9 5 ab ac bc ab 12 8 • (meta-associativity: m m = m m ) 7 11 Finitely presented. a a b a 1 32 4 • Divide and conquer proofs and computations. − − − − • “Algebraic Knot Theory”: If K is ribbon, 817 cl cl ab Z(K) ∈{cl2(Z): cl1(Z) = 1}. 1 2 Weaknesses, • m is non-linear. T c (Genus and crossing number a- T T • The product ωA is always Laurent, but proving this takes indu- re also deﬁnable properties). ction with exponentially many conditions. trivial ribbon example Theorem 1. ∃! an invariant γ: {pure framed S -component K bh(H; T). balloons / tails tangles} → R × M (R), where R = R = Z((T ) ) is the ring S ×S S a a∈S T of rational functions in S variables, intertwining “Simply- u v ω1ω2 S 1 S 2 knotted ω1 S 1 ω2 S 2 ⊔ simple R4 1. , −−−−−→ S 1 A1 0 , balloons ∞ S 1 A1 S 2 A2 embeddings 4 S 2 0 A2 and hoops” S ω a b S H ab ω c S a α β θ mc x y z / 2. −−−−−→  c γ + αδ/ ǫ + δθ/  , hoops heads b γ δ ǫ ≔1−β    S φ + αψ/ Ξ + ψθ/  Examples. x S φ ψ Ξ  Ta,Tb→Tc δ    1 a b  ǫx: x γ

1 a ±1 t " and satisfying |a; ! , −→ ; a 1 1 − T . a b b a  a 1 a  ǫu: u  b 0 T ±1   a  y   x In Addition, • This is really “just” a stitching  + − ρux: u ρux: u formula for Burau/Gassner [LD, KLW, CT]. x • L → ω is Alexander, mod units. “the generators” Shin Satoh • L → (ω, A) → ω det′(A − I)/(1 − T ′) is the Disturbing R2 R3 MVA, mod units. Conjecture = = • The “fastest” Alexander algorithm. VR3 • ? VR1 VR2 There are also formulas for strand deletion, K bh = u v = = = reversal, and doubling. M OC • Every step along the computation is the invariant of something. = = • Extends to and more naturally deﬁned on v/w-tangles. x CP CP • Fits in one column, including propaganda & implementation. UC Dictionary. = = Implementation key idea: ωεβ/Demo ↔ = (ω, A = (αab)) ↔ = “v-xing” ↔ (ω, λ = αabtahb) = ↔ ↔ blue is never “over” =: wK bh no! yet not OC: as UC: Dror Bar-Natan: Talks: Oberwolfach-1405: ωεβ≔http://www.math.toronto.edu/˜drorbn/Talks/Oberwolfach-1405/ Some very good formulas for the Alexander polynomial, 2 Operations Connected ⊔ Theorem 3 [BND, BN]. ∃! a homomorphic expansion, aka a ho- Punctures & Cuts Sums. momorphic universal ﬁnite type invariant Z of w-knotted balloons H uv and hoops. ζ ≔ log Z takes values in FL(T) × CW(T). If X is a space, π1(X) K: K tmw : ζ is computable! ζ of the Borromean tangle, to degree 5: is a group, π2(X) is an u v w Abelian group, and π + cyclic colour 1 permutations, acts on π2. for trees Proposition. The gene- x y x y rators generate.

xy ux K hmz : K tha : u v u v

z x y

Definition. lxu is the linking number of hoop x with balloon u. ≔ lxu Z Proposition [BN]. Modulo all re- For x ∈ H, σx u∈T Tu ∈ R = RT = ((Ta)a∈T ), the ring of rational functions in T variables. lations that universally hold for = − Theorem 2 [BNS]. ∃! an invariant β: wK bh(H; T) → R × the 2D non-Abelian Lie algebra and after some changes-of- MT×H(R), intertwining variable, ζ reduces to β and the [u, v] = cuv− cvu ω1ω2 H1 H2 ω1 H1 ω2 H2 ⊔ KBH operations on ζ reduce to the formulas in Theorem 2. 1. , −−−−−→ T1 A1 0 , T1 A1 T2 A2 T2 0 A2 A Big Question. Does it all extend to arbitrary 2-knots (not neces- ω H sarily “simple”)? To arbitrary codimension-2 knots? uv ω H 1 4 2 ∗ u α tmw BF Following [CR]. A ∈ Ω (M = R , g), B ∈ Ω (M, g ), 2. −−−−−→  w α + β  , v β   ≔  T Ξ  S (A, B) B, FA. T Ξ  Tu,Tv→Tw   M  xy  2 0 1 ∗ Rossi ω x y H hmz ω z H With κ: (S = R ) → M, β ∈ Ω (S, g), α ∈ Ω (S, g ), set 3. −−−−−→ , Ξ + Ξ T α β T α σxβ i ∗ O ≔ D D ∗ + ω x H νω x H (A, B,κ) β α exp β, dκ Aα κ B . thaux S 4. u α θ −−−−−→ u σxα/ν σxθ/ν , ν≔1+α The BF Feynman Rules. For an edge e, let Φe be its T φ Ξ T φ/ν Ξ − φθ/ν 3 1 direction, in S or S . Let ω3 and ω1 be volume forms β ± 1 x 1 1 x 3 Cattaneo and satisfying ǫx; ǫu; ρ −→ ; ; . on S and S 1. Then ux u u T ±1 − 1 u [D] ∗ ∗ Proposition. If T is a u-tangle and β(δT) = (ω, A), then ZBF = Φeω3 Φeω1 |Aut(D)| R2 R2 R4 R4 ab diagrams red black γ(T) = (ω, σ − A), where σ = diag(σa)a∈S . Under this, mc ↔ D e∈D e∈D ab ab ab S -vertices M-vertices tha tmc hmc . (modulo some STU- and IHX -like relations). References. [BN] D. Bar-Natan, Balloons and Hoops and their Universal Finite Type I- M 1 2 1 nvariant, BF Theory, and an Ultimate Alexander Invariant, ωεβ/KBH, 2 air 2 piece S S arXiv:1308.1721. 1 1 rattle S 2 1 1 2 [BND] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Ob- 2 1 0 S S S M 2 S jects I-II, ωεβ/WKO1, ωεβ/WKO2, arXiv:1405.1956, arXiv:1405.1955. 1 0 1 [BNS] D. Bar-Natan and S. Selmani, Meta-Monoids, Meta-Bicrossed Products, 1 1 S 1 0 1 2 1 0 2 1 0 and the Alexander Polynomial, J. of Knot Theory and its Ramifications 22-10 S S S 1 0 1 S 2 1 S (2013), arXiv:1302.5689. 2 0 S ground 0 piece S [CR] A. S. Cattaneo and C. A. Rossi, Wilson Surfaces and Higher Dimen- S 0 1 2 1 1 2 1 sional Knot Invariants, Commun. in Math. Phys. 256-3 (2005) 513–537, S S S degree = #(rattles) arXiv:math-ph/0210037. Issues. • Signs don’t quite work out, and BF seems to reproduce [CT] D. Cimasoni and V. Turaev, A Lagrangian Representation of Tangles, To- only “half” of the wheels invariant. pology 44 (2005) 747–767, arXiv:math.GT/0406269. • There are many more configuration space integrals than BF [KLW] P. Kirk, C. Livingston, and Z. Wang, The Gassner Representation for String Links, Comm. Cont. Math. 3 (2001) 87–136, arXiv:math/9806035. Feynman diagrams and than just trees and wheels. [LD] J. Y. Le Dimet, Enlacements d’Intervalles et Représentation de Gassner, • I don’t know how to define “finite type” for arbitrary 2-knots. Comment. Math. Helv. 67 (1992) 306–315. “God created the knots, all else in topology is the work of mortals.” Leopold Kronecker (modified) www.katlas.org Videos of all talks are available at their respective web pages Meta−Groups, Meta−Bicrossed−Products, and the Alexander Polynomial, 1 Dror Bar−Natan in Montreal, June 2013. http://www.math.toronto.edu/~ drorbn/Talks/Montreal-1306/ Abstract. I will define “meta-groups” and explain how one specificAlexander Issues. meta-group, which in itself is a “meta-bicrossed-product”, gives rise• Quick to compute, but computation departs from topology. to an “ultimate Alexander invariant” of tangles, that contains the• Extends to tangles, but at an exponential cost. Alexander polynomial (multivariable, if you wish), has extremely• Hard to categorify. good composition properties, is evaluated in a topologically mean- Idea. Given a group G and two “YB” ingful way, and is least-wasteful in a computational sense. If you ± ± pairs R± = (g±, g±) ∈ G2, map them Z go believe in categorification, that’s a wonderful playground. o u ± This work is closely related to work by Le Dimet (Com-to xings and “multiply along”, so that gu ment. Math. Helv. 67 (1992) 306–315), Kirk, Livingston and Wang (arXiv:math/9806035) and Cimasoni and Turaev + + + − − + + + + + + Z go gu go gu go gu go gu (arXiv:math.GT/0406269). − − gu go See also Dror Bar-Natan and Sam Selmani, Meta-Monoids, − − Meta-Bicrossed Products, and the Alexander Polynomial, arXiv:1302.5689. Sam Selmani ± ∓ ± ∓ This Fails! R2 implies that go go = e = gu gu and then R3 u v + + implies that go and gu commute, so the result is a simple 4D u v counting invariant. K A Group Computer. Given G, can store group elements and x y perform operations on them:

z xy x : g1 m z u : g2 xy ∗ u : g2 ...so that m u v : g3 xy v : g3 uz yz xu K hm mv = mu mv z y : g4 uz xy z : g1g4 (or mv ◦ mu = xu ◦ yz Riddle. People often w u v G{x,u,v,y} mv mu , in old- G{u,v,z} 1 study π1(X) = [S ,X] speak). 2 and π2(X) = [S ,X]. Also has Sx for inversion, ex for unit insertion, dx for register dele- z x Why not πT (X) := tion, ∆ for element cloning, ρ for renamings, and (D1,D2) 7→ x y x y xy y [T,X]? D1 ∪ D2 for merging, and many obvious composition axioms relat- b b ing those. 1 2 b P = {x : g ,y : g } ⇒ P = {dyP }∪{dxP } b uv th S1 S2 T K tmw K swux A Meta-Group. Is a similar “computer”, only its internal 817 structure is unknown to us. Namely it is a collection of sets 6 4 {Gγ } indexed by all finite sets γ, and a collection of opera- xy z x +6 8 ++10 + 9=1 tions mz , Sx, ex, dx, ∆ (sometimes), ρ , and ∪, satisfying 13 14 15 16 xy y 9 5 + + + the exact same linear properties. 112 2 8 γ 7 11 Example 0. The non-meta example, Gγ := G . 12 3 4 3 − − − − − − Example 1. Gγ := Mγ×γ(Z), with simultaneous row and 5 column operations, and “block diagonal” merges. Here if 7 x : a b “divide and conquer” P = then dyP = (x : a) and dxP = (y : d) so y : c d R1 R2 R3 x : a 0 = = = {dyP } ∪ {dxP } = 6= P . So this G is truly meta. y : 0 d ± A Standard Alexander Formula. Label the arcs 1 throughClaim. From a meta-group G and YB elements R ∈ G2 we (n + 1) = 1, make an n × n matrix as below, delete one rowcan construct a knot/tangle invariant. and one column, and compute the determinant: Bicrossed Products. If G = HT is a group presented as a b c a b c product of two of its subgroups, with H ∩ T = {e}, then also G = TH and G is determined by H, T , and the “swap” map + a c −1 1 − X X swth : (t, h) 7→ (h′,t′) defined by th = h′t′. The map sw b a b c c satisfies (1) and (2) below; conversely, if sw : T ×H → H ×T a− c −X X − 1 1 satisfies (1) and (2) (+ lesser conditions), then (3) defines a group structure on H × T , the “bicrossed product”.

t1 t2 h4 t1 t2 h4 11 2 2 H (1) (2) = = (3) 3 3 12 12 12 12 12 T tm1 sw14 =sw24 sw14 tm1 gm3 :=sw12 tm3 hm3 ′ ′ ′ ′ h1t1h2t2 = h1(h2t1)t2 = (h1h2)(t1t2)= h3t3 Videos of all talks are available at their respective web pages Meta−Groups, Meta−Bicrossed−Products, and the Alexander Polynomial, 2 A Meta-Bicrossed-Product is a collection of sets β(η,τ) andI mean business! uv xy th operations tmw , hmz and swux (and lesser ones), such that tm and hm are “associative” and (1) and (2) hold (+ lesser conditions). A meta-bicrossed-product deﬁnes a meta-group with Gγ := β(γ, γ) and gm as in (3). Example. Take β(η,τ) = Mτ×η(Z) with row operations for the tails, column operations for the heads, and a trivial swap. β Calculus. Let β(η,τ) be

ω h1 h2 · · · hj ∈ η, ti ∈ τ, and ω and  t1 α11 α12 · the αij are rational func-   ,  t2 α21 α22 · tions in a variable X with  .  . ω(1) = 1 and αij(1) = 0 (1) Some . · · · =   testing   ω · · · ω1 η1 ω2 η2 ω · · · ∪ τ1 α1 τ2 α2 uv tu α 2 tm : 7→ tw α + β , ω ω η η , 6 w tv β 1 2 1 2 . 5 . . γ = τ1 α1 0 4 1 . γ 3 τ2 0 α2 2 ω hx hy · · · ω hz · · · 4 1 hmxy : 7→ , 6 z . . 5 . α β γ . α + β + hαiβ γ 3 ... divide and conquer!

ω hx · · · ωǫ hx · · · 817 swth : tu α β 7→ tu α(1 + hγi/ǫ) β(1 + hγi/ǫ) , ux . . . γ δ . γ/ǫ δ − γβ/ǫ where ǫ := 1+ α and hci := i ci, and let P 1 ha hb 1 ha hb p m −1 Rab := ta 0 X − 1 Rab := ta 0 X − 1 . tb 0 0 tb 0 0 Theorem. Zβ is a tangle invariant (and more). Restricted to knots, the ω part is the Alexander polynomial. On braids, it 817, cont. is equivalent to the Burau representation. A variant for links contains the multivariable Alexander polynomial. Why Happy? • Applications to w-knots. • Everything that I know about the Alexander polynomial can be expressed cleanly in this language (even if without proof), except HF, but including genus, ribbonness, cabling, v-knots, knotted graphs, etc., and there’s potential for vast James generalizations. Waddell • The least wasteful “Alexander for tangles” Alexander I’m aware of. A Partial To Do List. 1. Where does it more • Every step along the computation is the in- simply come from? T variant of something. 2. Remove all the denominators. • Fits on one sheet, including implementation 3. How do determinants arise in this context? trivial & propaganda. 4. Understand links (“meta-conjugacy classes”). Further meta-monoids. Π (and variants), A (and quotients),5. Find the “reality condition”. T vT , ... 6. Do some “Algebraic Knot Theory”. −→ Further meta-bicrossed-products. Π (and variants), A (and7. Categorify. ribbon bh rbh quotients), M0, M, K , K , ... 8. Do the same in other natural quotients of the Meta-Lie-algebras. A (and quotients), S, ... v/w-story. −→ Meta-Lie-bialgebras. A (and quotients), . . . "God created the knots, all else in I don’t understand the relationship between gr and H, as it topology is the work of mortals." Leopold Kronecker (modified) example appears, for example, in braid theory. www.katlas.org Videos of all talks are available at their respective web pages

Trees and Wheels and Balloons and Hoops Kbh(T ; H). 15 Minutes on Topology Dror Bar-Natan, Zurich, September 2013 balloons / tails “Ribbon- ωεβ:=http://www.math.toronto.edu/~drorbn/Talks/Zurich-130919 T knotted u v 15 Minutes on Algebra balloons 4 Let T be a finite set of “tail labels” and H a finite set of and hoops” ribbon R ∞ embeddings “head labels”. Set S4 H M1/2(T ; H) := FL(T ) , H “H-labeled lists of elements of the degree-completed free Lie x y z hoops / heads algebra generated by T ”. Examples. x δ 1 anti-symmetry ǫx: x FL(T )= 2t2 − [t1, [t1,t2]] + . . . 2 Jacobi t ǫ : u . . . with the obvious bracket. u y u v u v v x Shin Satoh u + − x → y → − 22 ρux: u ρux: u M1/2(u, v; x,y)= λ = , 7 . . . x x y y “the generators” Operations M1/2 → M1/2. newspeak! uv More on δ δ Tail Multiply tmw is λ → λ (u, v → w), satisfies “meta- = uv uw vw uv δ associativity”, tmu tmu = tmv tmu . xy “v” Head Multiply hmz is λ → (λ\{x,y}) ∪ (z → bch(λx, λy)), satisfies R123, VR123, D, and no! where yet not [α,β] [α,[α,β]]+[[α,β],β] OC: as bch(α, β) := log(eαeβ)= α + β + + + . . . UC: 2 12 bh satisfies bch(bch(α, β), γ) = log(eαeβeγ ) = bch(α, bch(β, γ))• δ injects u-knots into K (likely u-tangles too). xy xz yz xy • δ maps v-tangles to Kbh; the kernel contains the above and and hence meta-associativity, hmx hmx = hmy hmx . conjecturally (Satoh), that’s all. Tail by Head Action thaux is λ → λ RCλx , where u • Allowing punctures and cuts, δ is onto. C−γ : FL → FL is the substitution u → e−γ ueγ, or more u Operations precisely, Connected ∗ Punctures & Cuts Sums. −γ − ad γ 1 Cu : u → e (u)= u − [γ, u]+ [γ, [γ, u]] − ..., uv 2 If X is a space, π1(X) K: K tmw : −β γ −γ −1 bch(α,β) α RCu β is a group, π2(X) u v w and RCu = (Cu ) . Then Cu = Cu Cu hence bch(α,β) β RCα is an Abelian group, RC = RCα RC u hence “meta uxy = (ux)y”, u u u and π acts on π . xy uz ux uy xy 1 2 hmz tha = tha tha hmz , uv −γ Riddle. People often x y x y uv γ tmw γ RCv γ uv and tmw Cw = Cu Cv tmw and hence “metastudy π (X) = [S1,X] x x x uv wx ux vx uv 1 newspeak! (uv) = u v ”, tm tha = tha tha tm . 2 w w and π2(X) = [S ,X]. xy ux K hmz : K tha : Wheels. Let M(T ; H) := M (T ; H) × CW(T ), whereWhy not πT (X) := 1/2 [T,X]? u v u v CW(T ) is the (completed graded) vector space of cyclic words b b b on T , or equaly well, on FL(T ): b 1 2 u v u v uv u v S S T z x y = − “Meta-Group-Action” v v ωεβ/antiq-ave Properties. uv xy • Associativities: mab mac = mbc mab, for m = tm,hm. Operations. On M(T ; H), define tmw and hmz as before, a a b a ux x x x uv wx ux vx uv and tha by adding some J-spice: • “(uv) = u v ”: tmw tha = tha tha tmw , (xy) x y xy uz ux uy xy λx • “u = (u ) ”: hmz tha = tha tha hmz . (λ; ω) → (λ, ω + Ju(λx)) RCu , 1 ab ab Tangle concatenations → π1 ⋉ π2. With dmc := tha where J (γ):= ds div (γ RCsγ) C−sγ, and ab ab u u u u tmc hmc , 0 u v u v u v Alekseev ab a bmc c u u u δ δ divu a bab c + dmc ab c γ Torossian Finite type invariants make − − Theorem Blue. All blue identities still hold. sense in the usual way, and Merge Operation. (λ1; ω1)∗(λ2; ω2) := (λ1 ∪ λ2; ω1 + ω2). “algebra” is (the primitive part of) “gr” of “topology”. Videos of all talks are available at their respective web pages Trees and Wheels and Balloons and Hoops: Why I Care Moral. To construct an M-valued invariant ζ of (v-)tangles,The β quotient is M divi- and nearly an invariant on Kbh, it is enough to declare ζ onded by all relations that uni- the generators, and verify the relations that δ satisfies. versally hold when when g is = − The Invariant ζ. Set ζ(ǫx) = (x → 0; 0), ζ(ǫu) = ((); 0), and the 2D non-Abelian Lie alge- x Q [u, v] = c v− c u u u bra. Let R = J{cu}u∈T K and u v ζ:u ; 0 u − ; 0 Lβ := R ⊗ T with central R and with [u, v] = cuv − cvu for x x x u, v ∈ T . Then FL → Lβ and CW → R. Under this, Theorem. ζ is (log of) the unique homomorphic universal finite type invariant on Kbh. → ((λx); ω) with λx = λuxux, λux,ω ∈ R, (. . . and is the tip of an iceberg) u∈T c + c ecu − 1 ecv − 1 Paper in progress with Dancso, ωεβ/wko u v cu bch(u, v) → c +c u + e v , e u v − 1 cu cv if γ = γvv then with cγ := γvcv,

cγ −1 cγ γ e − 1 cγ e − 1 u RCu = 1+ cuγu e u − cu γvv , cγ cγ v= u   c div γ = c γ , and J (γ) = log 1+ e γ −1 c γ , so ζ is u u u u cγ u u formula-computable to all orders! Can we simplify?

Repackaging. Given ((x → λux); ω), set cx := v cvλvx, cx e −1 ω cu replace λux → αux := cuλux c and ω → e , usetu = e , See also ωεβ/tenn, ωεβ/bonn, ωεβ/swiss, ωεβ/portfolio x and write αux as a matrix. Get “β calculus”. ζ is computable! ζ of the Borromean tangle, to degree 5: β Calculus. Let β(T ; H) be + cyclic colour permutations, ω x y ω and the αux’s are for trees  u α α  ux uy rational functions in  v α α  ,  vx vy variables tu, one for . With Selmani, . each u ∈ T . ωεβ/meta      ω ω1 H1 ω2 H2 ω ∗ u α T1 α1 T2 α2 uv tm : → w α + β , ω1ω2 H1 H2 , w v β . I have a nice free-Lie . . γ = T1 α1 0 calculator! . γ T2 0 α2 ω x y ω z xy hmz : . → . , . α β γ . α + β + αβ γ Tensorial Interpretation. Let g be a finite dimensional Lie algebra (any!). Then there’s τ : FL(T ) → Fun(⊕T g → g) ω x ωǫ x ux and τ : CW(T ) → Fun(⊕T g). Together, τ : M(T ; H) → tha : u α β → u α(1 + γ/ǫ) β(1 + γ/ǫ) , . . Fun(⊕T g →⊕H g), and hence . γ δ . γ/ǫ δ − γβ/ǫ τ ⊗H e : M(T ; H) → Fun(⊕T g →U (g)). where ǫ := 1+ α, α := v αv, and γ := v= u γv, and let ζ and BF Theory. (See Cattaneo-Rossi, + 1 x − 1 x arXiv:math-ph/0210037) Let A denote a g- Rux := Rux := −1 . 4 u tu − 1 u tu − 1 connection on S with curvature FA, and B a ∗ 4 On long knots, ω is the Alexander polynomial! g -valued 2-form on S . For a hoop γx, let Why happy? An ultimate Alexander inva- holγx (A) ∈ U(g) be the holonomy of A along γx. ∗ Cattaneo riant: Manifestly polynomial (time and si- For a ball γu, let Oγu (B) ∈ g be (roughly) the integral of B (transported via A to ∞) on γu. ze) extension of the (multivariable) Alexan- Loose Conjecture. For γ ∈ K(T ; H), der polynomial to tangles. Every step of the computation is the computation of the inva- R B∧FA Oγ (B)) τ DADBe e u holγ (A)= e (ζ(γ)). riant of some topological thing (no fishy Gaus- x u x sian elimination). If there should be an Alexander invariant That is, ζ is a complete evaluation of the BF TQFT. with a computable algebraic categorification, it is this one! See also /regina, /caen, /newton. “God created the knots, all else in ωεβ ωεβ ωεβ topology is the work of mortals.” May class: ωεβ/aarhus Class next year: ωεβ/1350 Leopold Kronecker (modified) www.katlas.org Paper: ωεβ/kbh Videos of all talks are available at their respective web pages

Dror Bar-Natan: Talks: Geneva-131024: ω :=http://www.math.toronto.edu/~drorbn/Talks/Geneva-131024 Finite Type Invariants of Ribbon Knotted Balloons and Hoops Abstract. On my September 17 Geneva talk (ω/sep) I de-Action 1. TC = scribed a certain trees-and-wheels-valued invariant ζ of ribbon knotted loops and 2-spheres in 4-space, and my October 8 + Geneva talk (ω/oct) describes its reduction to the Alexander ˜bh Q A = −→ polynomial. Today I will explain how that same invariant 4T = + arises completely naturally within the theory of finite type CP = 0 invariants of ribbon knotted loops and 2-spheres in 4-space. TH B degree=# of arrows = 0 HC balloons/tails d c = bh c d T = K (T ; H) (then connect using u v π : −→ a b a b xings or v-xings) = 0 ribbon R4 ζ −→ ∞ Deriving 4T . R3 embeddings 4 key: use S Start from = = + Goussarov-Polyak-Viro H FL H CW x y z (T ) × (T ) hoops/heads “trees” “wheels” ++ + = +++ in In/In+1 My goal is to tell you why such an invariant is expected, yet not to derive the computable formulas. ++= ++ R2 using TC Disturbing R3 Conjecture = = Action 2. d c c d VR3 a VR1 VR2 e bh u v = = = Z˜ : −→ = + + 1 + K = Q a b a b 2 M OC Exercise. = = R3. Prove prop- x −→ Satoh CP CP TC 4T TC erty U. = = UC Dictionary. = The Bracket-Rise Theorem. ↔ CP, B, = −−−→ −→ “v-xing” ↔ ST U, AS, bh ∼ −−−→ = A = IHX ↔ ↔ and blue is never “over” (2 in 1 out vertices) relations Expansions −−−→ −−−→ STU 1: = − STU2: = − the semi-virtual := − i.e.− or − n bh Let I := pictures with ≥ n semi-virts ⊂ K . −−−→ −−−→ STU − IHX − We seek an “expansion” 3 =TC: 0 = : = bh bh n n+1 bh Proof. Z : K → gr K = M I /I =: A 1 2 d − =2 1 = − satisfying “property U”: if γ ∈In, then n+1 Corollaries. (1) Related to Lie algebras! (2) Only trees and wheels Z(γ) = (0,..., 0, γ/I , ∗, ∗,...). X.-S. Lin persist. Why? • Just because, and this is vastly more general. Theorem. Abh is a bi-algebra. The space of its primitives is ⋆ H • Kbh/In+1 is “finite-type/polynomial invariants”. FL(T ) × CW(T ), and ζ = log Z. ∞ n • The Taylor example: Take K = C (R ), I =ζ is computable! ζ of the Borromean tangle, to degree 5: n n {f ∈ K: f(0) = 0}. Then I = {f : f vanishes like |x| } so + cyclic colour n n+1 permutations, I /I is homogeneous polynomials of degree n and Z is a for trees “Taylor expansion”! (So Taylor expansions are vastly more general than you’d think). ˜bh Plan. We’ll construct a graded A , a sur- A˜bh ˜bh bh z< O jective graded π : A → A , and a fil- Z˜ zz bh bh zz π Z˜ tered Z˜ : K →A so that π gr Z˜ = Id zz gr z (property U: if deg D = n, Z˜(π(D)) = bh bh K Z / A π(D) + (deg ≥ n)). Hence • π is an iso- I have a nice free-Lie morphism. • Z := Z˜ π is an expansion. calculator! “God created the knots, all else in topology is the work of mortals.” Leopold Kronecker (modified) www.katlas.org Videos of all talks are available at their respective web pages

Dror Bar-Natan: Talks: Bern-131104: ω :=http://www.math.toronto.edu/~drorbn/Talks/Bern-131104 The Kashiwara-Vergne Problem and Topology Video, handout, links at ω/ Abstract. I will describe a general machine, a close cousin 4D Knots. of Taylor’s theorem, whose inputs are topics in topology and whose outputs are problems in algebra. There are many inputs the machine can take, and many outputs it produces, but I will concentrate on just one input/output pair. When fed with a certain class of knotted 2-dimensional objects in A 4D knot by Carter and Saito ω/CS 4-dimensional space, it outputs the Kashiwara-Vergne Prob- lem (1978 ω/KV, solved Alekseev-Meinrenken 2006 ω/AM, elucidated Alekseev-Torossian 2008-2012 ω/AT), a problem about convolutions on Lie groups and Lie algebras. ω/F The Kashiwara-Vergne Conjecture. There exist two series F and G in the completed free Lie Dalvit Satoh algebra FL in generators x and y so that Kashiwara ω/Dal x+y−log eyex = (1−e− ad x)F +(ead y−1)G in FL The Generators “cup” and so that with z = log exey, Vergne

tr(ad x)∂xF + tr(ad y)∂yG in cyclic words → → = 1 ad x ad y ad z Alekseev = tr + − − 1 2 ead x − 1 ead y − 1 ead z − 1 “the crossing” ω/X + Implies the loosely-stated convolutions statement: Convolutions of invariant functions on a Meinrenken Lie group agree with convolutions of invariant →= → = “the + functions on its Lie algebra. Torossian vertex”

The Machine. Let G be a group, K = QG = { aigi : ai ∈ “v-xing” ω/vX Q, g ∈ G} its group-ring, I = { a g : a = 0} ⊂ K its i i i i P The Double Inflation Procedure. augmentation ideal. Let P P.S. (KP/Im+1)∗ is Vassiliev −∞ +∞ −∞ +∞ / finite-type / polynomial in- A = gr K := Im/Im+1. m≥0 variants. Note that A inheritsMd a product from G. Definition. A linear Z : K→A is an “expansion” if for any γ ∈ Im, Z(γ) = (0,..., 0, γ/Im+1, ∗,...), and a “homomor- → Riddle. phic expansion” if in addition it preserves the product. What band, inflated, Example. Let K = C∞(Rn) and I = {f : f(0) = 0}. Then gives the “Wen”? Im = {f : f vanishes like |x|m} so Im/Im+1 degree m ho-wKO. R2 R3 R4 = = = mogeneous polynomials and A = {power series}. The Taylor all wK :=PA types VR1 series is a homomorphic expansion! VR2 VR3 = = = K = The set of all Just for fun. = 2D projections “Planar of reality M CP CP Algebra”: = OC = = ω/Bear The objects = ··· UC K/K1 ←K/K2 ←K/K3 ←K/K4 ← are “tiles” that can be composed in Crop arbitrary planar ways to make bigger = = Rotate ··· tiles. Adjoin no! yet not OC: as An expansion Z is a choice of a UC: “progressive scan” algorithm. crop K/K1 ⊕⊕K1/K2 K2/K3⊕ K3/K4 ⊕⊕⊕K4/K5 K5/K6 ··· rotate → =→ = = = adjoin 3 R ker(K/K4→K/K3) The unary

In the finitely presented case, finding Z amounts to solving a w−operations Unzip along an annulus Unzip along a disk system of equations in a graded space. The Machine generalizes to arbitrary Theorem (with Zsuzsanna Dancso, ω/WKO). algebraic structures! There is a bijection between the set of homomor- ω/mac phic expansions for wK and the set of solutions “God created the knots, all else in of the Kashiwara-Vergne problem. This is the tip topology is the work of mortals.” of a major iceberg. Dancso, ω/ZD Leopold Kronecker (modified) www.katlas.org Videos of all talks are available at their respective web pages Dror Bar-Natan: Academic Pensieve: 2014-04: BF2C: http://drorbn.net/AcademicPensieve/2014-04/BF2C A Partial Reduction of BF Theory to Combinatorics, 1 continues http://www.math.toronto.edu/˜drorbn/Talks/Vienna-1402 Abstract. I will describe a semi-rigorous reduction of perturba- The BF Feynman Rules. For tive BF theory (Cattaneo-Rossi [CR]) to computable combina- an edge e, let Φe be its di- 3 1 torics, in the case of ribbon 2-links. Also, I will explain how rection, in S or S . Let ω3 and why my approach may or may not work in the non-ribbon and ω1 be volume forms on S 3 and S . Then for a 2-link case. Weak this result is, and at least partially already known 1 Cattaneo Rossi (Watanabe [Wa]). Yet in the ribbon case, the resulting invariant is (κt)t∈T , a universal finite type invariant, a gadget that significantly gener- [D] ∗ ∗ ζ = log Φeω3 Φeω1 |Aut(D)| R2 R2 R4 R4 alizes and clarifies the Alexander polynomial and that is closely diagramsX Z Z Z Z Yred Yblack D e∈D e∈D related to the Kashiwara-Vergne problem. I cannot rule out the S -vertices M-vertices possibility that the corresponding gadget in the non-ribbon case is an invariant in CW(FL|(T{z)) →}CW| {z(T)/∼}, “symmetrized cyclic will be as interesting. (good news in highlight) words in T”. M 1 A BF Feynman Diagram. BF Following [CR]. A ∈ Ω1(M = R4, g), B ∈ Ω2(M, g∗), 2 1 2 air 2 , ≔ , . piece S S S (A B) B FA 1 1 ZM rattle S 2 1 1 2 2 1 0 With κ: (S = R2) → M, β ∈ Ω0(S, g), α ∈ Ω1(S, g∗), set S S S M 2 S 1 0 1 i ∗ O(A, B,κ) ≔ DβDα exp β, d ∗ α + κ B . 1 S κ A 0 1 1 2 Z ZS ! 1 1 0 2 1 0 S S S 1 0 trees Decker Sets (“2D Gauss Codes”). 1 S 2 1 S 2 0 S snake 0 on wheels, S 0 S odd edges 1 1 2 1 1 2S S S degree = #(rattles) even rattles (only double curves are allowed in ribbon 2-knots) ~ ~ “a double curve” Ro1

~ Ro2 ~

~ ~ “a triple point” Ro3 Carter Banach Saito ~ ~ Ro4 Roseman [Ro] “a branch point” Some Examples. 4 ~ 3 ∞ Ro5 ~ 2 2 4 3 4 1 1 1 A 4D knot by Carter and Saito [CS] ~ ~ 2 Ro6 3 Dalvit A 4D knot by Dalvit [Da] ∞ “a w-knot” ~ ~ Ro7

∞ A 2-link

1 2 3 1 34 2 1 3 4 2 4 “ribbed cigar presentation” 1 3 4 A 2-twist spun trefoil by Carter- 2 Kamada-Saito [CKS]. Videos of all talks are available at their respective web pages Dror Bar-Natan: Academic Pensieve: 2014-04: BF2C: http://drorbn.net/AcademicPensieve/2014-04/BF2C A Partial Reduction of BF Theory to Combinatorics, 2 Theorem 1 (with Cattaneo, Dalvit (credit, no blame)). In the rib- Sketch of Proof. In 4D ax- ζ bon case, e can be computed as follows: ial gauge, only “drop down” red propagators, hence in the ribbon − case, no M-trivalent vertices. S integrals are ±1 k l m iff “ground pieces” run on nested curves as below, and exponentials arise when several propagators ζ e (+)k(−)l(+)m compete for the same double curve. And then the k!l!m! k,Xl,m≥0 combinatorics is obvious... + ++ +++ − 1 34 2 13 5 78 6 5 8 − k 2 4 6 7 m a ζ k m e (+) (−) a k!m! + + b kX,m≥0 Musings b + + + Chern-Simons. When the domain of BF is restricted to ribbon knots, and the target of Chern-Simons is restricted to trees and − − − wheels, they agree. Why? Is this all? What 2 2 Theorem 2. Using Gauss diagrams to represent knots and T- about the ∨-invariant? component pure tangles, the above formulas define an invariant (the “true” triple link- 1 1 → in CW(FL(T)) CW(T), “cyclic words in T”. ing number) = + • Agrees with BN-Dancso [BND] and with [BN2]. • In-practice computable! • Vanishes on braids. • Extends to w. • Contains Gnots. In 3D, a generic immersion of S 1 is an Alexander. • The “missing factor” in Levine’s factorization [Le] embedding, a knot. In 4D, a generic immersion (the rest of [Le] also fits, hence contains the MVA). • Related to of a surface has finitely-many double points (a / extends Farber’s [Fa]? • Should be summed and categorified. gnot?). Perhaps we should be studying these? References. Finite type. What are finite-type [Ar] V. I. Arnold, Topological Invariants of Plane Curves and Caustics, Uni- invariants for 2-knots? What − versity Lecture Series 5, American Mathematical Society 1994. would be “chord diagrams”? [BN1] D. Bar-Natan, Bracelets and the Goussarov filtration of the space of knots, Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Bubble-wrap-finite-type. Topology Monographs 4 1–12, arXiv:math.GT/0111267. There’s an alternative defini- [BN2] D. Bar-Natan, Balloons and Hoops and their Universal Fi- tion of finite type in 3D, due nite Type Invariant, BF Theory, and an Ultimate Alexander In- to Goussarov (see [BN1]). The Goussarov variant, http://www.math.toronto.edu/˜drorbn/papers/KBH/, arXiv:1308.1721. obvious parallel in 4D involves [BND] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W- “bubble wraps”. Is it any good? Knotted Objects: From Alexander to Kashiwara and Vergne, http://www.math.toronto.edu/˜drorbn/papers/WKO/. Shielded tangles. In 3D, one can’t zoom in and compute “the [CKS] J. S. Carter, S. Kamada, and M. Saito, Diagrammatic Computations for Chern-Simons invariant of a tangle”. Yet there are well-defined Quandles and Cocycle Knot Invariants, Contemp. Math. 318 (2003) 51–74. invariants of “shielded tangles”, and rules for their compositions. [CS] J. S. Carter and M. Saito, Knotted surfaces and their diagrams, Mathe- What would the 4D analog be? matical Surveys and Monographs 55, American Mathematical Society, Prov- idence 1998. [Da] E. Dalvit, http://science.unitn.it/˜dalvit/. [CR] A. S. Cattaneo and C. A. Rossi, Wilson Surfaces and Higher Dimen- sional Knot Invariants, Commun. in Math. Phys. 256-3 (2005) 513–537, “an associator” / arXiv:math-ph 0210037. Will the relationship with the Kashiwara-Vergne problem [BND] [Fa] M. Farber, Noncommutative Rational Functions and Boundary Links, Math. Ann. 293 (1992) 543–568. necessarily arise here? [Le] J. Levine, A Factorization of the Conway Polynomial, Comment. Math. Plane curves. Shouldn’t we understand integral / finite Helv. 74 (1999) 27–53, arXiv:q-alg/9711007. type invariants of plane curves, in the style of Arnold’s [Ro] D. Roseman, Reidemeister-Type Moves for Surfaces in Four-Dimensional + − Space, Knot Theory, Banach Center Publications 42 (1998) 347–380. J , J , and St [Ar], a bit better? Arnold [Wa] T. Watanabe, Configuration Space Integrals for Long n-Knots, the Alexan- der Polynomial and Knot Space Cohomology, Alg. and Geom. Top. 7 (2007) 47-92, arXiv:math/0609742. Continuing Joost Slingerland. . . http://youtu.be/YCA0VIExVhge “God created the knots, all else in topology is the work of mortals.” http://youtu.be/mHyTOcfF99o Leopold Kronecker (modified) www.katlas.org Convolutions on Lie Groups and Lie Algebras and Ribbon 2−Knots Disclaimer: "God created the knots, all else in Rough edges topology is the work of mortals." Dror Bar−Natan, Bonn August 2009, http://www.math.toronto.edu/~drorbn/Talks/Bonn−0908 remain! Leopold Kronecker (modified) The Bigger Picture... What are w-Trivalent Tangles? (PA :=Planar Algebra) knots =PA R123 : = , = , = Convolutions The Orbit &links statement Method 0 legs

trivalent Group-Algebra Subject =PA , R23, R4 : = = statement ﬂow chart tangles * →

Unitary wTT= statement Free Lie trivalent w- w- unary w- =PA statement w-tangles generators relations operations Algebraic statement The w−generators. Broken surface Alekseev Diagrammatic Torossian 2D Symbol statement statement

Knot-Theoretic True Dim. reduc. Alekseev, Toros- statement sian, Meinrenken Crossing Virtual crossing Movie Cap Wen www.math.toronto.edu/~drorbn/Talks/KSU−090407 W Vertices

Kashiwara Alekseev [1] smooth

Vergne Torossian singular R4 Homomorphic expansions for a ﬁltered algebraic structure K: A Ribbon 2-Knot is a surface S embedded in that bounds an immersed handlebody B, with only “ribbon singularities”;

opsÍK = K0 ⊃ K1 ⊃ K2 ⊃ K3 ⊃ . . . a ribbon singularity is a disk D of trasverse double points, ⇓ ↓ Z whose preimages in B are a disk D1 in the interior of B and

opsÍ gr K := K0/K1 ⊕ K1/K2 ⊕ K2/K3 ⊕ K3/K4 ⊕ . . . a disk D2 with D2 ∩ ∂B = ∂D2, modulo isotopies of S alone. An expansion is a ﬁltration respecting Z : K → gr K that “covers” the identity on gr K. A homomorphic expansion is = an expansion that respects all relevant “extra” operations. Filtered algebraic structures are cheap and plenty. In any The w-relations include R234, VR1234, M, Overcrossings 2 K, allow formal linear combinations, let K1 be the ideal Commute (OC) but not UC, W = 1, and funny interactions generated by diﬀerences (the “augmentation ideal”), and let between the wen and the cap and over- and under-crossings: m no! Km := h(K1) i (using all available “products”). yet not OC: as "An Algebraic Structure" UC: O = ψ1 •σ Challenge. O1 ψ2 O2 as Do the •1 Reidemeister! ψ4 W W but objects of ψ3 n o = O3 W kind 3 O4 W Reidemeister Winter

• Has kinds, objects, operations, and maybe constants. → = → = • Perhaps subject to some axioms.

• We always allow formal linear combinations. The unary

w−operations Unzip along an annulus Unzip along a disk Example: Pure Braids. PB is generated by x , “strand i n ij K = The set of all goes around strand j once”, modulo “Reidemeister moves”. Just for fun. = b/w 2D projections of reality An := gr PBn is generated by tij := xij − 1, modulo the 4T relations [tij ,tik + tjk] = 0 (and some lesser ones too). Much ··· happens in An, including the Drinfel’d theory of associators. K/K1 ←K/K2 ←K/K3 ←K/K4 ← Crop Our case(s). given a “Lie” Z: high algebra A := algebra g Rotate ··· K −−−−−−−−−−−−−→ −−−−−−−−−−→ “U(g)” Adjoin solving finitely many gr K low algebra: pictures represent equations in finitely An expansion Z is a choice of a many unknowns formulas “progressive scan” algorithm. K is knot theory or topology; gr K is finite combinatorics: crop K/K1 ⊕⊕K1/K2 K2/K3⊕ K3/K4 ⊕⊕⊕K4/K5 K5/K6 · · · bounded-complexity diagrams modulo simple relations. rotate = [1] http://qlink.queensu.ca/~4lb11/interesting.html 29/5/10, 8:42am adjoin = Also see http://www.math.toronto.edu/~drorbn/papers/WKO/ R ker(K/K4→K/K3) Convolutions on Lie Groups and Lie Algebras and Ribbon 2−Knots, Page 2 w Knot-Theoretic statement. There exists a homomorphic ex- From wTT to A . grm wTT := {m−cubes}/{(m+1)−cubes}: pansion Z for trivalent w-tangles. In particular, Z should

− − respect R4 and intertwine annulus and disk unzips: − forget Vassiliev

(1) ; topology

Polyak Goussarov w-Jacobi diagrams and A. Aw(Y ↑) =∼ Aw(↑↑↑) is (2)→ (3) → −→ 4T : + = +

= w VI: = + TC: Diagrammatic statement. Let R = exp Ë ∈ A (↑↑). There w w

Ï W exist ω ∈ A ( ) and V ∈ A (↑↑) so that STU: :=− := −

R =0 = +=0

deg= 1 #{vertices}=6 (1) = V 2 j R Diagrammatic to Algebraic. With (xi) and (ϕ ) dual bases of V ∗ k w R g and g and with [xi, xj ]= bij xk, we have A →U via k m

bji k P bkl V i j m l W W n

i j n l Penrose Cvitanovic ω ωω ϕ ϕ xn xm ϕ ϕ (2)unzip (3) unzip dim g W k m i j l bijbklϕ ϕ xnxmϕ ∈ U(Ig) i,j,k,l,m,n=1 V V X Unitary ⇐⇒ Algebraic. The key is to interpret Uˆ(Ig) as tan- Algebraic statement. With Ig := g∗ ⋊ g, with c : Uˆ(Ig) → gential differential operators on Fun(g): ∗ Uˆ(Ig)/Uˆ(g) = Sˆ(g∗) the obvious projection, with S the an- • ϕ ∈ g becomes a multiplication operator. tipode of Uˆ(Ig), with W the automorphism of Uˆ(Ig) induced • x ∈ g becomes a tangential derivation, in the direction of by flipping the sign of g∗, with r ∈ g∗ ⊗ g the identity element the action of ad x: (xϕ)(y) := ϕ([x, y]). ˆ ˆ ˆ ˆ ∗ and with R = er ∈ Uˆ(Ig) ⊗ Uˆ(g) there exist ω ∈ Sˆ(g∗) and • c : U(Ig) → U(Ig)/U(g)= S(g ) is “the constant term”. V ∈ Uˆ(Ig)⊗2 so that 2 x+y Unitary =⇒ Group-Algebra. ωx+ye φ(x)ψ(y) (1) V (∆ ⊗ 1)(R)= R13R23V in Uˆ(Ig)⊗2 ⊗ Uˆ(g) x+y ZZ x+y (2) V · SW V = 1 (3) (c ⊗ c)(V ∆(ω)) = ω ⊗ ω = ωx+y, ωx+ye φ(x)ψ(y) = V ωx+y,Ve φ(x)ψ(y)ωx+y x y x y G =hω ω ,e e V φ(x)ψ(y)ω i=hω ω ,e e φ(x)ψ(y)ω ω i Unitary statement. There exists ω ∈ Fun(g) and an (infinite x y x+ y x y x y order) tangential differential operator V defined on Fun(gx × 2 2 x y = ωxωye e φ(x)ψ(y). gy) so that ZZ [ (1) V ex+y = exeyV (allowing Uˆ(g)-valued functions) Convolutions and Group Algebras (ignoring all Jacobians). If ∗ G is finite, A is an algebra, τ : G → A is multiplicative then (2) V V = I (3) Vωx+y = ωxωy 2 G (Fun(G),⋆) ∼ (A, ·) via L : f 7→ f(a)τ(a). For Lie (G, g), Group-Algebrab statement.b There exists ω ∈ Fun(g) so that = τ =exp for every φ, ψ ∈ Fun(g)G (with small support), the following (g, +) ∋ x 0 S / x ˆ Fun(g) L0 / ˆ P e ∈ S(g)P S(g) holds in Uˆ(g): (shhh, ω2 = j1/2 ) PPP P expU PP −1 expG PP χ so Φ χ 2 x+y 2 2 x y PPP φ(x)ψ(y)ωx+ye = φ(x)ψ(y)ωxωye e . P( x τ1 / x L1 / gZZ×g gZZ×g (shhh, this is Duflo) (G, ·) ∋ e e ∈ Uˆ(g) Fun(G) Uˆ(g) x ˆ −1 x Convolutions statement (Kashiwara-Vergne). Convolutions of with L0ψ = ψ(x)e dx ∈ S(g) and L1Φ ψ = ψ(x)e ∈ −1 −1 invariant functions on a Lie group agree with convolutions Uˆ(g). Given ψi ∈ Fun(g) compare Φ (ψ1) ⋆ Φ (ψ2) and −1 R ˆ R of invariant functions on its Lie algebra. More accurately, Φ (ψ1 ⋆ ψ2) in U(g): (shhh, L0/1 are “Laplace transforms”) let G be a finite dimensional Lie group and let g be its Lie x y x+y algebra, let j : g → R be the Jacobian of the exponential ⋆ in G : ψ1(x)ψ2(y)e e ⋆ in g : ψ1(x)ψ2(y)e ZZ ZZ map exp : g → G, and let Φ : Fun(G) → Fun(g) be given We skipped... • The Alexander • v-Knots, quantum groups and by Φ(f)(x) := j1/2(x)f(exp x). Then if f,g ∈ Fun(G) are polynomial and Milnor numbers. Etingof-Kazhdan. Ad-invariant and supported near the identity, then • u-Knots, Alekseev-Torossian, • BF theory and the successful Φ(f) ⋆ Φ(g)=Φ(f⋆g). and Drinfel’d associators. religion of path integrals. • The simplest problem hyperbolic geometry solves.