KNOT POLYNOMIALS. ttp://knotebook.org (ITEP) September 21, ITEP, Moscow. September 21, / 73

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1 ITEP, Moscow September 21, 2015 ttp://knotebook.org (ITEP) September 21, / 73

2 H K R (q A) A=q N = Tr R P exp A K SU(N) September 21, / 73

3 S = κ 4π d 3 x Tr (AdA + 2 ) 3 A3 q = exp 2πi κ + N September 21, / 73

4 Can be lifted to extended polynomials H B R{q p k } which depend on infinitely many time-variables {p k } and on particular braid representation B of the oriented knot K. September 21, / 73

5 H K R (q A) H B R{q p k} p k = Ak A k q k q k At this topological locus the r.h.s. does not depend on the choice of B for a given K. September 21, / 73

6 Character expansion separates dependencies on the braid and the time variables: HR{q p B k } = crq(q)s B Q {p k } Q m R Sum is over Young diagrams Q of the size Q = m R September 21, / 73

7 More artful character expansions (especially in the Hall-Littlewood or MacDonald polynomials) can be even more useful for particular applications September 21, / 73

8 An m-strand braid is parameterized by a sequence of integers a (j) i with i = 1, 2,..., m 1 and j = 1,..., n a (1) 1 = 2, a (1) 2 = 2, a (2) 1 = 1, a (2) 2 = 3 (the closure is the knot 8 10 ) September 21, / 73

9 crq B (q) are represented as traces in the spaces of intertwining operators of combinations of simple matrices, c B RQ(q) = Tr Q j ( m 1 i=1 R a(j) i i,i+1 ) September 21, / 73

10 Representation is parameterized by the Young diagram: an ordered set of l(r) positive integers R = {r 1 r 2... r l(r) > 0} = [r 1, r 2,..., r l(r) ] September 21,

11 Find and investigate the equations, which the HOMFLY and extended HOMFLY polynomials satisfy as functions of a (j) i and r k. September 21,

12 Definition of knot polynomials Evaluation of knot polynomials Properties of knot polynomials September 21,

13 Definitions/evaluation: Perturbative expansion in 1/k leads to Vassiliev invariants and to Vogel s algebra of triple-ended diagrams (related to Konnes-Kreimer formalism) September 21,

14 Gauge choices quadratic actions (free field representations): A z = 0: A0 A propagator 1 z Kontsevich integral A 0 = 0: ɛab A a Ȧ b sign(t)δ (2) (z) 2d knot diagrams September 21,

15 Knot diagrams: Integrable lattice theory on arbitrary graph Reidemeister invariance = YB relation R-matrix calculus Conformal blocks Hypercube method (good also for virtual knots and for superpolynomials) September 21,

16 R-matrix approach: c B RQ(q) = Tr Q j ( m 1 i=1 Skein relations Û i ˆRa ij Û i ) September 21,

17 R-matrix, acting in the channel [1] [1] = [2] [11] has two eigenvalues, q and 1/q, associated with the two irreducible representations [2] and [11] respectively, and thus satisfies the Hecke algebra constraint ( ) R q )(R + q 1 = 0 September 21,

18 Skein relations in fund.rep.: H (... a ij+1...) H (... a ij 1...) = = ( q q 1) H (... a ij...) September 21,

19 For bigger representations R, the product R R = S S and R-matrix in this channel has many different eigenvalues: as many as there are irreducible representations S in this expansion. September 21,

20 These eigenvalues are equal to ɛ S q κ S, where κ S is an eigenvalue of the cut-and-join operator: Ŵ [2], associated with its eigenfunction character S S {p k }, and ɛ S = ±1, depending on R and S. Therefore, this R-matrix satisfies ( ) R ɛ S q κ S = 0 S September 21,

21 S ( ) e / a ij ɛ S q κ S H (...a ij...) R = 0 for any variable a ij. September 21,

22 For the first symmetric representation )( ) (R q )(R 6 + q 2 R 1 = 0 H (... a ij+3...) [2] (1+q 2 +q 6 )H (... a ij+2...) [2] + +q 2 (1+q 4 +q 6 )H (... a ij+1...) [2] q 8 H (... a ij...) [2] = September 21,

23 Cabling procedure to evaluate colored polynomials: H K R m = H K m R where K m denotes m-cabling of the knot K September 21,

24 HOMFLY for the l.h.s. can be presented as a sum over the irreducible representations, H K R m = S H K S To get HS K, one calculates HK m R in rep.r and then project the result onto irrep Q. September 21,

25 ( ) R q )(R + q 1 = 0 = P [2] = 1 + qr 1 + q 2 P [11] = q2 qr 1 + q 2 September 21,

26 [1] [1] [1] = [3] + 2[21] + [111] R 3 = R 12 R 13 )( ) (R 3 q )(R 2 3 q 2 R 2 3+R 3 +1 = 0 September 21,

27 P [3] = (1 q2 R 3 )(R R 3 + 1) (1 q 4 )(1 + q 2 + q 4 ) P [111] = q 6(R 3 q 2 )(R R 3 + 1) (1 q 4 )(1 + q 2 + q 4 ) September 21,

28 P ± [21] = ±e±πi/6 (q 2 R 3 1)(R 3 q 2 ) 3 (R 3 e ±2πi/3 ) 1 + q 2 + q 4 September 21,

29 Mixing matrixes U R 3 = R 12 R 13 = RURU R i,i+1 = sum over paths in rep.graph September 21,

30 Representation graph [1] [2] [11] [3] [21] [111] [4] [31] [22] [211] [1111]... September 21,

31 The multiplicity M Q of the representation Q in Q is obviously equal to the number of directed paths in the representation graph, connecting and Q. More generally, M RQ is equal to the number of directed paths between R and Q. September 21,

32 The matrices ˆR i,i+1, i = 1,..., m 1 can be represented in the basis of paths between Q and and they have extremely simple form in this basis. September 21,

33 First of all, with each index i of the matrix ˆR i,i+1 one associates a level i in the graph. A given path P is passing through exactly one vertex P i at level i and through some two adjacent vertices P i 1 and P i+1 at levels i 1 and i + 1. September 21,

34 The structure of the representation graph is such that these P i 1 and P i+1 are connected either by a single two-segment path (singlet) (then it is a fragment of our P) or by two such paths (doublet), the segments of our path P and another path P. September 21,

35 In the former case (singlet) our path P provides a diagonal element in ˆR (i) and it is equal to either q or 1/q. In the language of Young diagrams the singlet appears when the two boxes added to the diagram P i 1 in order to form P i+1 lie either in the same row, then we put q at the diagonal of ˆRi ; or in the same column, then we put 1/q. September 21,

36 September 21,

37 September 21,

38 In the latter case (the doublet) the two boxes are neither in the same row nor in the same column, and the two paths P and P form a 2 2 block in ˆR (i). This block is described as follows. September 21,

39 First, that of the paths P and P which lies to the left of the other, corresponds to the left column and to the first row of the 2 2 block. September 21,

40 Second, the Young diagrams P i+1 is obtained by adding two boxes to the diagram P i 1, and the two paths correspond to doing this in two different orders, thus providing at the intermediate stage the two adjacent vertices P i and P i. September 21,

41 The two added boxes are connected by a hook in the Young diagram, which has length n (measured between the centers of the two boxes). September 21,

42 q n c n s n s n q n c n c n = 1 = q q 1 [n] q q n q n s n = 1 c 2 n = [n 1]q [n + 1] q [n] q September 21,

43 7 n = 7 [5521] [6521] [5522] [6522] September 21,

44 ALGEBRAIC PROPERTIES September 21,

45 Special polynomials: H K R (q = 1 A) = ( ) R σ (A) K representations R September 21,

46 exp ( H K R (q A) = ( σ K (A)) R z +l( ) 2 S ( A z 2 ) ϕ R ( ) ϕ R ( ) = symmetric-group characters Hurwitz τ-function ) September 21,

47 DIFFERENTIAL EXPANSION: H K r (A, q 2 ) = 1+ {A/q} r s=1 [r]! [s]![r s]! G K s (A, q) s 1 {Aq r+j } j=0 {x} = x x 1 and [n] = {q n }/{q} September 21,

48 For generic knot Gs K is a non-factorizable Laurent polynomial of A and q, but for some knots it can be further factorized. September 21,

49 What is important, if Gs K is divisible by some differential {Aq k }, the same is true for all other Gs K with s > s. This property allows one to introduce defect functions νs K and µ K s = s 1 νs K : September 21,

50 G K s (A, q) = F K s (A, q) ν K s 1 j=0 {Aq j } = = F K s (A, q) s 2 µ K s j=0 {Aq j } September 21,

51 which are both(!) monotonically increasing function of s, ν K s ν K s, µ K s µ K s for s < s i.e. both grow but not faster than s. September 21,

52 For A = q N with any fixed N, positive or negative, F s (q N, q) {q} µk s (1) i.e. at fixed N the s-the term of differential expansion is actually of the order {q} 2s. September 21,

53 It turns out that νs K as a function of s has a very special shape, fully parameterized by a single integer δ K 1, which we call the defect of differential expansion: defect δ K = 1 = µ K s = s 2, ν K s = September 21, K K K

54 s : µ K s s s September 21,

55 s : µ K s 2s 3 s September 21,

56 GENERALIZATIONS: H K r+m = H K r H K m q 2m =1 September 21,

57 H K R+M? = H K m H K R q 2m =1 connecte provided both R and R + M are Young diagrams. The following picture is an explanation of what we mean by R + M: September 21,

58 connected M of unit width discon September 21,

59 Kashaev polynomial: K K R (A) = H K R ( ) q 2 = e 2πi/ R, A is the value of colored HOMFLY at a primitive root of unity q 2 R = 1. September 21,

60 For all single hook diagrams R Kashaev polynomial is easily expressed through the special polynomial: K K R (A) = K K [1] (A R ) = H K [1] R = [r, 1 k ] (q 2 = 1, A R ) September 21,

61 Dual to a property of colored Alexander polynomials: H K r (A = 1, q) = H K 1 (A = 1, q r ) R = [r, 1 k ] September 21,

62 When A moves away from 1 concentric circles deform and collide with Alexander circles (we plot zeros of H 10 ): September 21,

63 A = 1 A = September 21,

64 A = A = September 21,

65 We can also consider perturbation in purely imaginary direction A = 1 A = 1 i September 21,

66 A = 1 i 10 6 A = 1 i September 21,

67 For the trefoil all the roots of Alexander polynomial are unimodular. A = 1 A = September 21,

68 A = A = September 21,

69 and A = 1 A = 1 i September 21,

70 A = 1 i 10 2 A = 1 i September 21,

71 Zeroes of resultant q 2(H 4 1 1, H4 1 k ) for the figure eight knot K = 4 1 : September 21,

72 September 21,

73 September 21,

74 THANKS FOR YOUR KIND ATTENTION! September 21,