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Matrix Integrals and Knot Theory 1 Matrix Integrals and Knot Theory 1 Matrix Integrals and Knot Theory Paul Zinn-Justin et Jean-Bernard Zuber math-ph/9904019 (Discr. Math. 2001) math-ph/0002020 (J.Kn.Th.Ramif. 2000) P. Z.-J. (math-ph/9910010, math-ph/0106005) J. L. Jacobsen & P. Z.-J. (math-ph/0102015, math-ph/0104009). math-ph/0303049 (J.Kn.Th.Ramif. 2004) G. Schaeffer and P. Z.-J., math-ph/0304034 P. Z.-J. and J.-B. Z., review article in preparation Carghjese, March 25, 2009 Matrix Integrals and Knot Theory 2 Courtesy of “U Rasaghiu” Matrix Integrals and Knot Theory 3 Main idea : Use combinatorial tools of Quantum Field Theory in Knot Theory Plan I Knot Theory : a few definitions II Matrix integrals and Link diagrams 1 2 4 dMeNtr( 2 M + gM ) N N matrices, N ∞ − × → RemovalsR of redundancies reproduces recent results of Sundberg & Thistlethwaite (1998) ⇒ based on Tutte (1963) III Virtual knots and links : counting and invariants Matrix Integrals and Knot Theory 4 Basics of Knot Theory a knot , links and tangles Equivalence up to isotopy Problem : Count topologically inequivalent knots, links and tangles Represent knots etc by their planar projection with minimal number of over/under-crossings Theorem Two projections represent the same knot/link iff they may be transformed into one another by a sequence of Reidemeister moves : ; ; Matrix Integrals and Knot Theory 5 Avoid redundancies by keeping only prime links (i.e. which cannot be factored) Consider the subclass of alternating knots, links and tangles, in which one meets alternatingly over- and under-crossings. For n 8 (resp. 6) crossings, there are knots (links) which cannot be drawn ≥ in an alternating form. Asymptotically, the alternating are subdominant. Major result (Tait (1898), Menasco & Thistlethwaite, (1991)) Two alternating reduced knots or links represent the same object iff they are related by a sequence of “flypes” on tangles Problem Count alternating prime links and tangles Matrix Integrals and Knot Theory 6 A 8-crossing non-alternating knot Matrix Integrals and Knot Theory 7 Matrix Feynman diagrams and link diagrams Consider integral over complex (non Hermitean) matrices p n q m g MMi l + N[ t trMM† + tr(MM†)2] i l j k dM e − 2 j k R oriented (double) lines in propagators and vertices. ⇒ When N ∞, leading contribution from genus zero (“planar”) diagrams : → 1 gn ∞ ∑ limN 2 logZ = planar diagrams symm.factor → N with n vertices 2 0 for example, to second order N N Matrix Integrals and Knot Theory 8 Moreover : Conservation of arrows alternating diagram ! ⇒ But going from complex matrices to hermitian matrices doesn’t affect the planar limit . up to a global factor 2. Moral After removing redundancies (incl. flypes), counting of Feynman diagrams of M4 integral, (over hermitian matrices) N t trM2 g trM4 Z = dM e [ 2 + 4 ] Z − for N ∞, yields the counting of alternating links and tangles. → Matrix Integrals and Knot Theory 9 Non perturbative results on M4 integral, N ∞ → N[ t trM2 + g trM4] Compute large N limit of integral Z = dM e − 2 4 by saddle point method, or orthogonal polynomials,orR . In the N ∞ limit, continuous distribution of eigenvalues λ with density → u(λ) of support [ 2a,2a] (deformed semi-circle law) − 1 g g u(λ)= (1 2 a2 λ2) 4a2 λ2 2π − t2 − t2 − g p 3 a4 a2 + 1 = 0 t2 − Thus “planar” limit of trM4 integral 1 Z(t,g) 1 1 lim log = F(t,g)= loga2 (a2 1)(9 a2) N ∞ N2 Z(t,0) 2 − 24 − − → Matrix Integrals and Knot Theory 10 p 3g (2p 1)! ∞ p 7/2 F(t,g) = ∑ 2 − As p Fp const(12) p− p=1 t p!(p + 2)! → ∼ ... 1 2 2 Also 2-point function G2 = 3t a (4 a )= − and (connected and “truncated”) 4-point function 2 2 (5 2a )(a 1) Γ = = − − (4 a2)2 − Matrix Integrals and Knot Theory 11 Counting Links and Tangles For the knot interpretation of previous counting, many irrelevant diagrams have to be discarded. “Nugatory” and “non-prime” are removed by adjusting t = t(g) so that = 1 (“wave function renormalisation”). Matrix Integrals and Knot Theory 12 In that way, correct counting of links . .up to 6 crossings ! ... (a) 2 3 4 42 5 5 52 21 1 1 1 1 2 1 (b) ... 1 1 1 1 1 1 1 4 3 2 4 5 ~ (c) ~ p 7/2 Asymptotic behaviour Fp const(27/4) p ∼ − Matrix Integrals and Knot Theory 13 What happens for n 6 crossings ? Flypes ! ≥ ~ Must quotient by the flype equivalence ! Original combinatorial treatment (Sundberg & Thistlethwaite, Z-J&Z) rephrased and simplified by P. Z.-J. : it amounts to a coupling constant renormalisation g g0 ! In other words, 1 2 g0 4 → start from Ntr tM M , fix t = t(g0) as before. Then compute Γ(g0) 2 − 4 and determine g g as the solution of 0( ) 2 g0 = g 1 + , − (1 g)(1 + Γ(g0)) − then the desired generating function is Γ˜ (g)= Γ(g0). Matrix Integrals and Knot Theory 14 Indeed let H(g) be the generating function of “horizontally-two- + + + ... H = = particle-irreducible” diagrams (cannot separate the left part from the right by cutting two lines) and then write Γ=H/(1 H) − ... Matrix Integrals and Knot Theory 15 ~ Γ˜ ˜ But thus, with , H denoting generating functions of flype equivalence classes of prime tangles, resp 2PI tangles and if ˜ ˜ ˜ Γ˜ Γ˜ H′ H = g + H′, = g + g + ˜ 1 H′ − ... Matrix Integrals and Knot Theory 16 Return to Γ(g0) Γ(g ) ... 0 suggests to determine g0 = g0(g) by demanding that g0 = g 2gH˜ − ′ g 0 Three relations between g0, H˜ ′, g and Γ˜ (g) Eliminating H˜ ′ and then Matrix Integrals and Knot Theory 17 Eliminating g0 gives Γ˜ (g)= Γ(g0(g)), the generating function of (flype-equivalence classes of) tangles. Find Γ˜ = g + 2g2 + 4g3 + 10g4 + 29g5 + 98g6 + 372g7 + 1538g8 + 6755g9 + p ··· 101+√21001 5/2 Asymptotic behaviour Γ˜ p const p ∼ 40 − All this reproduces the results of Sundberg & Thistlethwaite . ⋆ Can we go further ? Control the number of connected components ? i.e. count knots rather than links ? Matrix Integrals and Knot Theory 18 Coloured Links and Tangles N tr 1 ∑n M2 + g ∑n M M M M (N) ∏n 2 a=1 a 4 a,b=1 a b a b Z (n,g)= a=1 dMa e − R Each connected component may come in n colours ∑∞ k If we write the free energy F(n,g)= k=1 Fk(g)n , Fk = generating function of diagrams with k loops. In particular, F1(g), that of knots. Unfortunately this is computable only for n = 2,1,2 − [P.Z.-J. 99, Z-J–Z 00] ⋆ Open and important problem to understand such integrals in the n 0 → limit (replicas, combinatorics. .) Matrix Integrals and Knot Theory 19 Another direction : Virtual Links Higher genus contributions to matrix integral What do they count ? Suggested that knots/links live on other manifolds Σh I × Virtual knots and links [Kauffman] : equivalence classes of 4-valent diagrams with ordinary under- or over-crossings plus a new type of virtual crossing, Matrix Integrals and Knot Theory 20 Equivalence w.r.t. generalized Reidemeister moves Matrix Integrals and Knot Theory 21 From a different standpoint : Virtual knots (or links) seen as drawn in a “thickened” Riemann surface ΣΣ := Σ [0,1], modulo isotopy in ΣΣ, and × modulo orientation-preserving homeomorphisms of Σ, and addition or subtraction of empty handles. But this is precisely what Feynman diagrams of the matrix integral do for us ! Thus return to the integral over complex matrices N t trMM† g tr MM† 2 Z(g,N)= dM e [ + 2 ( ) ] Z − and compute F(g,t,N)= logZ beyond the leading large N limit ... Matrix Integrals and Knot Theory 22 ∞ 2 2h (h) F(g,t,N)= ∑ N − F (g,t) h=0 F(h)(g) : Feynman diagrams of genus h F(1) computed by Morris (1991) F(2) and F(3) by Akermann and by Adamietz (ca. 1997) As before, determine t = t(g,N) so as to remove the non prime diagrams. Find the generating function of tangle diagrams Γ(g)= 2∂F/∂g 2 − Γ(0)(g) =g+2g2+6g3+22g4+91g5+408g6+1938g7+9614g8+49335g9+260130g10+O(g11) Γ(1)(g) =g+8g2+59g3+420g4+2940g5+20384g6+140479g7+964184g8+6598481g9+45059872g10+O(g11 Γ(2)(g) =17g3+456g4+7728g5+104762g6+1240518g7+13406796g8+135637190g9+1305368592g10+O(g11 Γ(3)(g) =1259g5+62072g6+1740158g7+36316872g8+627368680g9+9484251920g10+O(g11) Γ(4)(g) =200589 g7+14910216 g8+600547192 g9+17347802824g10+O(g11) Γ(5)(g) =54766516 g9+5554165536g10+O(g11) Matrix Integrals and Knot Theory 23 1 1 The genus 0 and 1 2-crossing alternating virtual link diagrams in the two representations, the Feynman diagrams on the left, the virtual diagrams on the right : for each, the inverse of the weight in F is indicated Matrix Integrals and Knot Theory 24 6 6 2 2 6 6 order 3, genus 0 and 1 Matrix Integrals and Knot Theory 25 8 8 2 order 4, genus 0 Matrix Integrals and Knot Theory 26 8 2 2 2 2 2 2 2 1 4 4 4 1 1 order 4, genus 1 Matrix Integrals and Knot Theory 27 8 1 1 order 4, genus 2 Matrix Integrals and Knot Theory 28 Removing the flype redundancies. ~ ~ First occurences of flype equivalence in tangles with 3 crossings Matrix Integrals and Knot Theory 29 Removing the flype redundancies. ~ ~ First occurences of flype equivalence in tangles with 3 crossings Matrix Integrals and Knot Theory 30 It is suggested that it is (necessary and) sufficient to quotient by the planar flypes, thus to perform the same renormalization g g0(g) as for genus 0.
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