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T Polynomial Time Knot Polynomials, A Abstrant. The value of things is inversely correlated with their Dror Bar-Natan: Talks: Iowa-1603: ωεβ≔http://drorbn.net/Iowa-1603/ computational complexity. “Real time” machines, such as our Work in Progress on Polynomial Time Knot Polynomials, A brains, mostly run linear time algorithms, and there’s stillalot we don’t know. Anything we learn about things doable in linear time Meta-Associativity Runs. is truly valuable. Polynomial time we can in-practice run, even if we have to wait; these things are still valuable. Exponential time we can play with, but just a little, and exponential things must be beautiful or philosophically compelling to deserve attention. Values further diminish and the aesthetic-or-philosophical bar further rises as we go further slower, or un-computable, or ZFC-style intrin- R3 ... divide and conquer! sically infinite, or large-cardinalish, or beyond. 2 I will explain some things I know about polynomial time knot polynomials and 6 explain where there’s more, within reach. 5 4 1 3 (v-)Tangles. ⊔ T1 T2 T1 T2 2 , 1 4 6 5 T a b c 3 ab (meta-associativity: mc 1 4 T T mab mxc = mbc max) 2 3 x y x y +6 ++10 + 13 14 15 16 Why Tangles? 9 5 TτT κ T 12 8 • 7 11 Finitely presented. 1 32 4 • Divide and conquer computations. − − − − U ∈ Tn 1 ∈ An τ τ 8 • “Alg. Knot Theory”: If K is ribbon, z 17 z(K) ∈{κ(ζ): τ(ζ) = 1}. T2n A2n Closed Components. The Halacheva meta-trace trc satisfies κ κ mab tr = mba tr and computes the MVA for all links in the (Genus and crossing number ∈ T ∈ A c c c c ribbon K 1 z(K) 1 atlas, but its domain is not understood: are also definable properties). Faster is better, leaner is meaner! ω c S 1T 2 trc µω S Theorem 1. ∃! an invariant z0 : {pure framed S -component c α θ −−−−−−−−−−→ ≔ Z S Ξ + ψθ/µ tangles} → Γ0(S ) R × MS ×S (R), where R = RS = ((Ta)a∈S ) is S ψ Ξ µ ≔ 1 − α the ring of rational functions in S variables, intertwining Halacheva ω ω S S ω S ω S ⊔ 1 2 1 2 1 1 , 2 2 −−−−−→ S A 0 , S A S A 1 1 1 1 2 2 ! S A 2 0 2 τ: trivial example ω a b S κ: ribbon mab µω c S a α β θ c Weaknesses. • mab and tr are non-linear. • The product ωA is −−−−−−−−−−−−→ c γ + αδ/µ ǫ + δθ/µ , c c b γ δ ǫ T , T → T always Laurent, but my current proof takes induction with expo- a b c S φ + αψ/µ Ξ + ψθ/µ S φ ψ Ξ µ ≔ 1 − β nentially many conditions. • I still don’t understand tr , “unita- c 1 a b rity”, the algebra for ribbon knots. Where does it come from? z0 1 a ±1 " and satisfying | ; ! , −→ ; a 1 1 − T . a a b b a a 1 a v-Tangles. R2 R3 M b 0 T ±1 = = = a vT ≔PA In Addition • The matrix part is just a stitching VR1 = VR2 VR3 formula for Burau/Gassner [LD, KLW, CT]. = = • K 7→ ω is Alexander, mod units. ′ ′ R2 R3 • L 7→ (ω, A) 7→ ω det (A − I)/(1 − T ) is the flying =MM =CA = = MVA, mod units. pogs v n n+1 ⊗S G • The fastest Alexander algorithm I know. Let I ≔ h» − i. Then A ≔ I /I =“universal U(Dg) ”= • There are also formulas for strand deletion, M. Polyak & T. Ohtsuki − Q = + (Also IHX) reversal, and doubling. @ Heian Shrine, Kyoto • Every step along the computation is the invariant of something. − = − = • Extends to and more naturally defined on v/w-tangles. Fine print: No sources no sinks, AS vertices, internally acyclic, deg = (#vertices)/2. • Fits in one column, including propaganda & implementation. Likely Theorem. [EK, En] There exists a homomorphic expan- Implementation key idea: ωεβ/Demo sion (universal finite type invariant) Z : vT → Av. (issues suppressed) (ω, A = (αab)) ↔ Too hard! Let’s look for “meta-monoid” quotients. (ω, λ = αabtahb) The w Quotient P = 0 Aw U(FL(S )S ⋉ CW(S )) Dror Bar-Natan: Talks: Iowa-1603: Work in Progress on ωεβ≔http://drorbn.net/Iowa-1603/ Polynomial Time Knot Polynomials, B Theorem 2 [BND]. ∃! a homomorphic expansion, aka a ho- Definition. (Compare [BNS, BN]) A The Abstract Context momorphic universal finite type invariant Zw of pure w-tangles. meta-monoid is a functor M : (finite sets, zw ≔ log Zw takes values in FL(S )S × CW(S ). injections)→(sets) (think “M(S ) is quantum GS ”, for G a group) z is computable. z of the Borromean tangle, to degree 5 [BN]: along with natural operations ∗: M(S 1) × M(S 2) → M(S 1 ⊔ S 2) ab + cyclic colour whenever S 1 ∩ S 2 = ∅ and mc : M(S ) → M((S \{a, b}) ⊔{c}) permutations, , ∈ < \{ } for trees whenever a b S and c S a, b , such that ab xc bc ax meta-associativity: mx my = mx my ab de de ab meta-locality: mc m f = m f mc ,({and, with ǫb = M(S ֒→ S ⊔{b ab ba meta-unit: ǫb ma = Id = ǫb ma . Claim. Pure virtual tangles PvT form a meta-monoid. Theorem. S 7→ Γ0(S ) is a meta-monoid and z0 : PvT → Γ0 is a morphism of meta-monoids. Theorem. There exists an extension of Γ0 to a bigger meta- monoid Γ01(S ) = Γ0(S ) × Γ1(S ), along with an extension of z0 to z : PvT → Γ , with (I have a fancy free-Lie calculator!) (ωεβ/FLD) Nice, but too hard! 01 01 Γ (S ) = R ⊕ V ⊕ V⊗2 ⊕ V⊗3 ⊕ S2(V)⊗2 (with V ≔ R hS i). Proposition [BN]. Modulo all re- 1 S S Furthermore, upon reducing to a single variable everything is lations that universally hold for b b the 2D non-Abelian Lie alge- = − polynomial size and polynomial time. ab bra and after some changes-of- Furthermore, Γ01 is given using a “meta-2-cocycle ρc over Γ0”: ab ab ab w In addition to m → m , there are R -linear m : Γ (S ⊔ variable, z reduces to z . [u, v] = buv− bvu c 0c S 1c 1 0 ab Back to v – the 2D “Jones Quotient”. {a, b}) → Γ1(S ⊔{c}), a meta-right-action α : Γ1(S ) × Γ0(S ) → c Λ Γ1(S ) RS -linear in the first variable, and a first order differential ab Λ Λ Λ Λ operator (over RS ) ρc : Γ0(S ⊔{a, b}) → Γ1(S ⊔{c}) such that b b = = = (ζ , ζ ) mab = ζ mab, (ζ , ζ ) αab mab + ζ ρab b c Λ c 0 1 c 0 0c 1 0 1c 0 c V. Jones Λ What’s done? The braid part, with still-ugly formulas. Contains the Jones and “swinging” ...yet What’s missing? A lot of concept- and detail-sensitive work tow- Alexander polynomials, still too hard! ab ab ab ards m1c, α , and ρc . The “ribbon element”. The OneCo Quotient. Likely related to [ADO] example = 0, only one co-bracket is allowed. Everything should work, and everything is being worked! References. a ribbon singularity a clasp singularity (Eppstein) [ADO] Y. Akutsu, T. Deguchi, and T. Ohtsuki, Invariants of Colored Links, J. A bit about ribbon knots. A “ribbon knot” is a knot that can be of Knot Theory and its Ramifications 1-2 (1992) 161–184. [BN] D. Bar-Natan, Balloons and Hoops and their Universal Finite Type I- presented as the boundary of a disk that has “ribbon singulari- nvariant, BF Theory, and an Ultimate Alexander Invariant, ωεβ/KBH, ties”, but no “clasp singularities”. A “slice knot” is a knot in arXiv:1308.1721. S 3 = ∂B4 which is the boundary of a non-singular disk in B4. [BND] D. Bar-Natan and Z. Dancso, Finite Type Invariants of W-Knotted Ob- Every ribbon knots is clearly slice, yet, jects I-II, ωεβ/WKO1, ωεβ/WKO2, arXiv:1405.1956, arXiv:1405.1955. Conjecture. Some slice knots are not ribbon. [BNS] D. Bar-Natan and S. Selmani, Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial, J. of Knot Theory and its Ramifications 22-10 Fox-Milnor. The Alexander polynomial of a ribbon knot is alw- (2013), arXiv:1302.5689. ays of the form A(t) = f (t) f (1/t). (also for slice) [CT] D. Cimasoni and V. Turaev, A Lagrangian Representation of Tangles, To- pology 44 (2005) 747–767, arXiv:math.GT/0406269. [En] B. Enriquez, A Cohomological Construction of Quantization Functors of Lie Bialgebras, Adv. in Math. 197-2 (2005) 430-479, arXiv:math/0212325. [EK] P. Etingof and D. Kazhdan, Quantization of Lie Bialgebras, I, Selecta Mathematica 2 (1996) 1–41, arXiv:q-alg/9506005. [GST] R. E. Gompf, M. Scharlemann, and A. Thompson, Fibered Knots and Potential Counterexamples to the Property 2R and Slice-Ribbon Conjectures, Geom. and Top. 14 (2010) 2305–2347, arXiv:1103.1601. [KLW] P. Kirk, C. Livingston, and Z. Wang, The Gassner Representation for ]: a slice knot that might not String Links, Comm. Cont. Math. 3 (2001) 87–136, arXiv:math/9806035. be ribbon (48 crossings). [LD] J. Y. Le Dimet, Enlacements d’Intervalles et Repr´esentation de Gassner, GST Comment. Math. Helv. 67 (1992) 306–315. [ “God created the knots, all else in topology is the work of mortals.” Help Needed!I’m slow and feeble-minded. Leopold Kronecker (modified) www.katlas.org Dror Bar-Natan: Talks: Iowa-1603: Work in Progress on ωεβ≔http://drorbn.net/Iowa-1603/ Polynomial Time Knot Polynomials, C http://drorbn.net/AcademicPensieve/2012-01/one/The_Most_Important_Missing_Infrastructure_Project_in_Knot_Theory.pdf Dror Bar-Natan: Academic Pensieve: 2012-01: The Most Important Missing Infrastructure Project in Knot Theory January-23-12 10:12 AM An "infrastructure project" is hard (and sometimes non-glorious) work that's done now and pays off later.
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