Matrix Integrals and Knot Theory 1

Transcription

1 Matrix Integrals and Knot Theory 1 Matrix Integrals and Knot Theory Paul Zinn-Justin and Jean-Bernard Zuber math-ph/ (Discr. Math. 2001) math-ph/ (J.Kn.Th.Ramif. 2000) P. Z.-J. (math-ph/ , math-ph/ ) J. L. Jacobsen & P. Z.-J. (math-ph/ , math-ph/ )... math-ph/ (J.Kn.Th.Ramif. 2004) G. Schaeffer et P. Z.-J., math-ph/ Trieste, 13 May 2005

2 Matrix Integrals and Knot Theory 2 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 R dm e Ntr( 1 2 M2 + gm 4 ) N N matrices, N Removals of redundancies reproduces recent results of Sundberg & Thistlethwaite (1998) based on Tutte (1963) III New! Virtual knots and links : counting and invariants

3 Matrix Integrals and Knot Theory 3 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 : ; ;

4 Matrix Integrals and Knot Theory 4 Avoid redundancies by keeping only prime links (i.e. which cannot be factored) Classification of the prime knots (see atlas), counting of links and tangles #cr #knots #links (8) (14) 8 21 (18) 50 (39) 9 49 (41) 132 (96) ,388,705 (379,799)? (1,016,975)

5 Matrix Integrals and Knot Theory 5

6 Matrix Integrals and Knot Theory 6 Consider the subclass of alternating knots, links and tangles, in which one meets alternatingly over- and under-crossings.

7 Matrix Integrals and Knot Theory 7 Courtesy of Prof. P. Dorey

8 Matrix Integrals and Knot Theory 8 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

9 Matrix Integrals and Knot Theory 9 A 8-crossing non-alternating knot

10 Matrix Integrals and Knot Theory 10 Matrix Integrals: Feynman Rules N N Hermitean matrices M, dm = i dm ii i< j drem i j dimm i j Z = R dme N[ 1 2 trm2 + g 4 trm4 ] Feynman rules: propagator i j 4-valent vertex : p q i j n m k l = gnδ jk δ lm δ np δ qi l k = 1 N δ ilδ jk For each connected diagram contributing to logz: fill each closed index loop with a disk discretized closed 2-surface Σ

11 Matrix Integrals and Knot Theory 11 Count powers of N in a connected diagram each vertex N; each double line N 1 ; each loop N. Thus N #vert. #lines+#loops = N χ Euler(Σ) [ t Hooft (1974)] logz = conn. surf.σ N 2 2genus(Σ) g #vert.(σ) symm. factor

12 Matrix Integrals and Knot Theory 12 Matrix Feynman diagrams and link diagrams Consider integral over complex (non Hermitean) matrices i j l k M M + R dm e N[ t trmm + g 2 tr(mm ) 2 ] oriented (double) lines in propagators and vertices. p q i j n m k l When N, leading contribution from genus zero ( planar ) diagrams: lim N 1 N 2 logz = planar diagrams with n vertices g n symm.factor gn 2 gn 0

13 Matrix Integrals and Knot Theory 13 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 M 4 integral, (over hermitian matrices) Z = Z dm e N[ t 2 trm2 + g 4 trm4 ] for N, yields the counting of alternating links and tangles.

14 Matrix Integrals and Knot Theory 14 Non perturbative results on M 4 integral, N Write integral Z = R dm e N[ t 2 trm2 + g 4 trm4 ] in terms of eigenvalues λi Z(t, g) = Z N i=1 dλ i exp{2 i< j ( λ log λ i λ j N 2 t i i 2 gλ4 i 4 ) } In the N limit, continuous distribution of λ with density u(λ) of support [ 2a,2a] (deformed semi-circle law) u(λ) = 1 2π (1 2 g t 2 a2 g t 2 λ2 ) 4a 2 λ 2 3 g t 2 a4 a = 0

15 Matrix Integrals and Knot Theory 15 Thus planar limit of trm 4 integral lim N 1 Z(t,g) log N2 Z(t,0) = F(t,g) = 1 2 loga (a2 1)(9 a 2 ) = ( 3g (2p 1)! )p As p F p=1 t2 p const(12) p p 7/2 p!(p + 2)!... Also 2-point function G 2 = 3t 1 a2 (4 a 2 ) = and (connected and truncated ) 4-point function Γ = = (5 2a2 )(a 2 1) (4 a 2 ) 2

16 Matrix Integrals and Knot Theory 16 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).

17 Matrix Integrals and Knot Theory 17 In that way, correct counting of links...up to 6 crossings! (a) (b) (c) ~ ~ Asymptotic behaviour F p const(27/4) p p 7/2

18 Matrix Integrals and Knot Theory 18 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 g 0! In other words, start from Ntr ( 1 2 tm2 g 0 4 M 4), fix t = t(g 0 ) as before. Then compute Γ(g 0 ) and determine g 0 (g) as the solution of ( ) 2 g 0 = g 1 +, (1 g)(1 + Γ(g 0 )) then the desired generating function is Γ(g) = Γ(g 0 ).

19 Matrix Integrals and Knot Theory 19 Indeed let H(g) be the generating function of horizontally-twoparticle-irreducible diagrams (cannot separate the left part from the right by cutting two lines) H = = + + and then write Γ=H/(1 H) $ %!! & ' "" ##

20 Matrix Integrals and Knot Theory 20 )+ ), )( )* )/)/ )0)0 )- ). But ~ )( )* )+ ), )- ). )/)/ )0)0 thus, with Γ, H denoting generating functions of flype equivalence classes of prime tangles, resp 2PI tangles and if H = g + H, Γ = g + g Γ + H 1 H 2D 2E C2C B2B C2C B2B C2C ; 2: 2; 2: 2; = 2< 2= 2< 2= 2? 2> 2? 2> 2? @ 2A 2@ 2A...

21 Matrix Integrals and Knot Theory 21 Return to Γ(g 0 ) GMGM GJGJ GL GNGN Γ( g ) GIGI GJGJ GK GL... 0 GIGI GJGJ GK GL GFGF GHGH suggests to determine g 0 = g 0 (g) by demanding that g 0 = g 2g H g 0 RPR SPS RPR SPS POPOPO PQPQPQ Three relations between g 0, H, g and Γ(g) Eliminating H and then

22 Matrix Integrals and Knot Theory 22 Eliminating g 0 gives Γ(g) = Γ(g 0 (g)), the generating function of (flype-equivalence classes of) tangles. Find Γ = g + 2g 2 + 4g g g g g g g 9 + ) p Asymptotic behaviour Γ p const p 5/2 ( Note that 12 > 27 4 = 6.75 > as it should. All this reproduces the results of Sundberg & Thistlethwaite. Can we go further? Control the number of connected components?

23 Matrix Integrals and Knot Theory 23 New Direction : Virtual Links Higher genus contributions to matrix integral What do they count? Suggested that knots/links on other manifolds Σ g 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,

24 Matrix Integrals and Knot Theory 24 Equivalence w.r.t. generalized Reidemeister moves

25 Matrix Integrals and Knot Theory 25 From a different standpoint : Virtual knots (or links) seen as drawn in a thickened Riemann surface Σ := Σ [0, 1], modulo isotopy in Σ, 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 Z(g,N) = Z dm e N[ t trmm + g 2 tr(mm ) 2 ] and compute F(g,t, N) = log Z beyond the leading large N limit...

26 Matrix Integrals and Knot Theory 26 F(g,t,N) = F (h) (g) : Feynman diagrams of genus h N 2 2h F (h) (g,t) h=0 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+2g 2 +6g 3 +22g 4 +91g g g g g g 10 +O(g 11 ) Γ (1) (g) =g+8g 2 +59g g g g g g g g 10 +O( Γ (2) (g) =17g g g g g g g g 10 +O Γ (3) (g) =1259g g g g g g 10 +O(g 11 ) Γ (4) (g) = g g g g 10 +O(g 11 ) Γ (5) (g) = g g 10 +O(g 11 )

27 Matrix Integrals and Knot Theory 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

28 Matrix Integrals and Knot Theory order 3, genus 0 and 1

29 Matrix Integrals and Knot Theory order 4, genus 0 2

30 Matrix Integrals and Knot Theory order 4, genus 1

31 Matrix Integrals and Knot Theory order 4, genus 2

32 Matrix Integrals and Knot Theory 32 Removing the flype redundancies. ~ ~ First occurences of flype equivalence in tangles with 3 crossings

33 Matrix Integrals and Knot Theory 33 It is suggested that it is (necessary and) sufficient to quotient by the planar flypes, thus to perform the same renormalization g g 0 (g) as for genus 0. Generalized flype conjecture: For a given (minimal) genus h, Γ (h) (g) = Γ (h) (g 0 ) is the generating function of flype-equivalence classes of virtual alternating tangles. Then asymptotic behavior ( # inequivalent tangles of order p = Γ (h) p ) p p 5 2 (h 1). Test this generalized flype conjecture by computing invariants of virtual links 1 linking numbers 2 polynomials : Jones, cabled Jones, Kauffman,... 3 Alexander-Conway polynomials and their multi-variable extensions... 4 fundamental group π

34 Matrix Integrals and Knot Theory 34 Up to order 4 (4 real crossings), this suffices to distinguish all flype-equivalence classes: Conjecture Higher orders: sometimes difficult to distinguish images under discrete symmetries (mirror, global flip =mirror under-cr over-cr.)?... Examples: A genus-1 order-5 virtual diagram which is distinguished from its mirror image through the 2-cabled Jones polynomial

35 Matrix Integrals and Knot Theory 35 At order 5, a pair of virtual flipped knots of genus 1, distinguished by their Alexander-Conway polynomial.

36 Matrix Integrals and Knot Theory 36 A pair of virtual flipped knots of genus 1, conjectured to be non equivalent. A pair of virtual flipped knots of genus 2, conjectured to be non equivalent.

37 Matrix Integrals and Knot Theory 37 Conclusions Field theoretic methods : Feynman diagrams and matrix integrals, but also transfer matrix methods, offer new and powerful ways of handling the counting of links/tangles. Some progress, but still many open issues. Count knots (rather than links)? K p = # knots with p crossings. Consider a k-colouring of links, then term linear in k...? Asymptotic behaviour of K p as p? K p Cτ p p γ 3, γ = [G. Schaeffer and P. Z.-J.] Non alternating Links and Knots?????????????