Patterns and Stability in the Coefficients of the Colored Jones Polynomial. Katie Walsh Advisor: Justin Roberts

Transcription

1 Patterns and Stability in the Coefficients of the Colored Jones Polynomial Advisor: Justin Roberts

2 The Middle Coefficients of the Colored Jones Polynomial 1 The Middle Coefficients of the Colored Jones Polynomial Jones Polynomial 2 Introduction 3 Definitions Hyperbolic Volume Conjecture 4 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence

3 The Middle Coefficients of the Colored Jones Polynomial

4 The Middle Coefficients of the Colored Jones Polynomial 5

5 The Middle Coefficients of the Colored Jones Polynomial The 5 th colored Jones Polynomial for figure 8 knot is: 1 q 20 1 q 19 1 q q 15 1 q 14 1 q 13 1 q 12 1 q q 10 1 q 9 2 q 8 2 q 7 1 q q 5 1 q 4 2 q 3 2 q 2 1 q +7 q 2q2 2q 3 q 4 +6q 5 q 6 2q 7 2q 8 q 9 +5q 10 This has coefficients: q 11 q 12 q 13 q q 15 q 18 q 19 + q 20 {1, 1, 1, 0, 0, 3, 1, 1, 1, 1, 5, 1, 2, 2, 1, 6, 1, 2, 2, 1, 7, 1, 2, 2, 1, 6, 1, 2, 2, 1, 5, 1, 1, 1, 1, 3, 0, 0, 1, 1, 1}

6 The Middle Coefficients of the Colored Jones Polynomial {1, 1, 1, 0, 0, 3, 1, 1, 1, 1, 5, 1, 2, 2, 1, 6, 1, 2, 2, 1, 7, 1, 2, 2, 1, 6, 1, 2, 2, 1, 5, 1, 1, 1, 1, 3, 0, 0, 1, 1, 1} We can plot these: Figure: Coefficients of the 5 th Colored Jones Polynomial for the Figure Eight Knot

7 The Middle Coefficients of the Colored Jones Polynomial Figure: Coefficients of the 20 th Colored Jones Polynomial for the Figure Eight Knot

8 The Middle Coefficients of the Colored Jones Polynomial Figure: Coefficients of the 50 th Colored Jones Polynomial for the Figure Eight Knot

9 The Middle Coefficients of the Colored Jones Polynomial Figure: Coefficients of the 95 th Colored Jones Polynomial for the Figure Eight Knot

10 The Middle Coefficients of the Colored Jones Polynomial Figure: Coefficients of the 95 th Colored Jones Polynomial for the Figure Eight Knot Divided by Sin

11 The Middle Coefficients of the Colored Jones Polynomial Constant Coefficient of the Colored Jones Polynomial of the Figure 8 Knot Constant Coefficient Number of Colors

12 The Middle Coefficients of the Colored Jones Polynomial Normalized Growth Rate of the Constant Term ln(constant Coef)*2 /N Number of Colors

13 The Middle Coefficients of the Colored Jones Polynomial Knot Knot Diagram Volume Trefoil (3 1 ) Not Hyperbolic Figure Eight (4 1 ) Not Hyperbolic Table: Hyperbolic Volumes of Different Knots

14 The Middle Coefficients of the Colored Jones Polynomial 1 In the middle, the coefficients of J K,N are approximately periodic with period N. 2 There is a sine wave like oscillation with an increasing amplitude on the first and last quarter of the coefficients. 3 We can see that the oscillation persists throughout the entire polynomial. The amplitude starts small, grow steadily and then levels off in the middle and then goes back down in a similar manner.

15 The Middle Coefficients of the Colored Jones Polynomial

16 Introduction The Jones Polynomial Definition A knot is an embedding f : S 1 S 3. A knot is usually represented through projection into R 2 such that: At most two segments come together at any one point Whenever two segments meet we designate which arc is the over crossing and which is the under crossing. Figure: Five Knots. Are any of them the same?

17 Introduction The Jones Polynomial Definition ([9]) Two knots are equivalent if there is an orientation preserving piecewise linear homeomorphism h : S 3 S 3 that maps one knot to the other. Figure: There are three different knot types in this figure. The red knots are unknots, the green knots are trefoils and the blue knot is a figure 8 knot.

18 Introduction The Jones Polynomial We can use knot invariants to help us tell whether or not two knot diagrams represent equivalent knots. Definition A knot invariant is a property of a knot that does not change under ambient isotopy. If two knots have different values for any knot invariant, then it is impossible to transform one into the other, thus they are not equivalent.

19 Introduction The Jones Polynomial Theorem (Reidemeister 1928) Any two equivalent knots are related by planar isotopy and a sequence of the three Reidemeister moves. Reidemeister 1: Reidemeister 2: Reidemeister 3:

20 Introduction The Jones Polynomial Definition The Kauffman Bracket is an invariant of framed knots. It is characterized by the skein relation below. = 1 D = ( A 2 A 2 ) D = A + A 1

21 Introduction The Jones Polynomial Reidemeister 2: Reidemeister 3:

22 Introduction The Jones Polynomial Reidemeister 2: Reidemeister 3: Reidemeister 1:

23 Introduction The Jones Polynomial We can adapt the Kauffman Bracket to be a knot invariant. Definition The Jones Polynomial of a knot is a knot invariant of a knot K with diagram D defined by ( ) V (K) = ( A) 3w(D) D where w(d) is the writhe of the diagram. q 1/2 =A 2 w(d) = # #

24 Introduction The Jones Polynomial Knot Knot Diagram Jones Polynomial Trefoil (3 1 ) q + q 3 q 4 Figure Eight (4 1 ) q 2 q q + q q 2 + q 4 q 5 + q 6 q q 2 + q 4 q 5 + q 6 q 7 Table: Jones Polynomials of Different Knots

25 Introduction The Jones Polynomial Knot Knot Diagram Jones Polynomial Trefoil (3 1 ) q + q 3 q 4 Mirror Image(3 1 ) q 1 + q 3 q 4 Figure Eight (4 1 ) q 2 q q + q 2 Mirror Image (4 1 ) q 2 q q 1 + q 2 Table: Jones Polynomials of Knot and their Mirror Images

26 Introduction The Jones Polynomial Knot Knot Diagram Jones Polynomial 5 1 q 2 + q 4 q 5 + q 6 q q 2 + q 4 q 5 + q 6 q q 2 + q 4 q 5 + q 6 q 7 Table: Jones Polynomials of Different Knots

27 Definitions Hyperbolic Volume Conjecture We can generalize the Jones polynomial to the colored Jones polynomials. The colored Jones polynomial assigns to each knot a family of Laurent polynomials, indexed by N, the color.

28 Definitions Hyperbolic Volume Conjecture Knot Knot Diagram Colored Jones Polynomial (2+1 dim rep) Trefoil (3 1 ) q 2 + q 5 q 7 + q 8 q 9 q 10 + q 11 Figure Eight (4 1 ) q 6 q 5 q 4 + 2q 3 q 2 q+ 3 q 1 + q q 3 q 4 q 5 + q q 4 + q 7 q 9 + q 10 q 12 + q 13 2q 15 + q 16 q 18 + q q q 1 3q 2 + q 3 + 3q 4 4q 5 + 2q 6 + 2q 7 3q 8 + 2q 9 + q 10 3q q 12 2q q 15 q 16 q q 18 q 19 q 20 + q 21 Table: Colored Jones Polynomials of Different Knots

29 Definitions Hyperbolic Volume Conjecture The N dimensional colored Jones polynomial is also a linear combination of the Jones polynomial on cablings of the knots. We can express this linear combination recursively as: For example, g 3 = z 2 1 so, g 1 = 1 g 2 = z g i = zg i 1 g i 2. J 3,41 = V ( ) 1

30 Definitions Hyperbolic Volume Conjecture We have formulas for the figure eight knot, twist knots, K p and (1, 2p 1, r 1) pretzel knots, K p,r. p full twists 2p-1 r-1 The twist knot K p The (1, 2p 1, r 1) pretzel knot

31 Definitions Hyperbolic Volume Conjecture Theorem (Habiro and Le) J N,4 1 (a 2 ) = N 1 n=0 {N n}{n 1 + 1} {N + n} {N} where {n} = a n a n and {n}! = {n}{n 1} {1}.

32 Definitions Hyperbolic Volume Conjecture Theorem (Habiro and Le) For a twist knots with p twists, where J N,K p (a 2 ) = N 1 n=0 f Kp,n = a n(n+3)/2 1 (a a 1 ) n As standard, and {N n}{n 1 + 1} {N + n} f Kp,n {N} n ( 1) k µ p 2k [2k+1] [n]! [n + k + 1]![n k]! k=0 q = a 2, a = A 2, {n} = a n a n, [n] = an a n µ i = ( 1) i A i 2 +2i a a 1

33 Definitions Hyperbolic Volume Conjecture Theorem (W.) A pretzel knot of the form K p,l = P(1, 2p 1, l 1) has the colored Jones polynomial J N,K p,l (a 2 ) = N 1 n=0 ( 1) n [ ] N+n (a a 1 ) 2n c N n 1 n,p When l is even this reduces to {2n+1}!{n}! {1} [N] nk=0 ( 1) k(l+1) [2k+1]µ l/2 2k [n+k+1]![n k]!. J n,k p,l (a 2 ) = N 1 n=0 ( 1)n[ ] N+n N n 1 c {2n+1}! n,p [N] {1} c n,l/2. Here c n,p = 1 (a a 1) n n ( 1) k µ p [n]! 2k [2k + 1] [n + k + 1]![n k]!. k=0

34 Definitions Hyperbolic Volume Conjecture Knot Twists Pretzel Notation (p,l) (1,3,0) or (1,1,1) (2,1) or (1,2) 4 1 (1,1,2) (1,3) 5 1 (1,5,0) (3,1) (1,3,1) or (1,1,3) (2,2) or (1,4) 6 1 (1,1,4) (1,5) 6 2 (1,3,2) (2,3) 7 1 (1,7,0) (4,1) (1,1,5) or (1,5,1) (1,6) or (3,2) 7 4 (1,3,3) (2,4) 8 1 (1,1,6) (1,7) 8 2 (1,5,2) (3,3) 8 4 (1,3,4) (2,5)

35 Definitions Hyperbolic Volume Conjecture Definition The hyperbolic volume of a hyperbolic knot K is the volume of the unique hyperbolic metric on the knot complement (S 3 \ K) We can calculate the hyperbolic volume of the knot by building its complement out of ideal tetrahedrons. The hyperbolic volume of a knot is a knot invariant.

36 Definitions Hyperbolic Volume Conjecture Knot Knot Diagram Volume Trefoil (3 1 ) Not Hyperbolic Figure Eight (4 1 ) Not Hyperbolic Table: Hyperbolic Volumes of Different Knots

37 Definitions Hyperbolic Volume Conjecture Conjecture (Kashaev, Murakami, Marakami) The Hyperbolic Volume Conjecture states that: vol(s 3 log J N,K \ K) = 2π lim (e2πi/n ) N N The hyperbolic volume conjecture has been proved for: torus knots, the figure-eight knot, Whitehead doubles of (2, p)-torus knots, positive iterated torus knots, Borromean rings, (twisted) Whitehead links, Borromean double of the figure-eight knot, Whitehead chains, and fully augmented links (see [11]).

38 Definitions Hyperbolic Volume Conjecture What does the head and the tail tell us about the geometry of the knot? Theorem (Dasbach, Lin) Volume-ish Theorem: For an alternating, prime, non-torus knot K let J K,2 (q) = a n q n + + a m q m be the Jones polynomial of K. Then 2v 0 (max( a m 1, a n+1 ) 1) Vol(S 3 K) Vol(S 3 K) 10v 0 ( a n+1 + a m 1 1). Here, v is the volume of and ideal regular hyperbolic tetrahedron.

39 Definitions Hyperbolic Volume Conjecture Some research areas related to the coefficients of the colored Jones polynomial: Head and Tail of the Colored Jones Polynomial The Middle Coefficients (The Belly?) Higher Order Stability and Asymptotic Behavior

40 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence

41 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence

42 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence N Highest Terms of the Colored Jones Polynomial of q 2 q + 1 q 1 + q 2 3 q 6 q 5 q 4 + 2q 3 q 2 q + 3 q 1 q q 12 q 11 q 10 +0q 9 + 2q 8 2q 6 + 3q 4 3q q 20 q 19 q 18 +0q 17 +0q q 15 q 14 q q 30 q 29 q 28 +0q 27 +0q 26 +q q q q 42 q 41 q 40 +0q 39 +0q 38 +q 37 +0q q q 56 q 55 q 54 +0q 53 +0q 52 +q 51 +0q 50 +q 49 +

43 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence The Head of the Colored Jones Polynomial of 4 1 J N,4 1 (q) = N 1 n=0 k=1 J N,4 1 (q) HT = n {N k}{n + k} N 1 k =1 (1 q k ) Theorem (Euler s Pentagonal Number Theorem) (1 x n ) = n=1 k= ( 1) k x k(3k 1)/2 = 1 x x 2 + x 5 + x 7 x 12 x 15 +

44 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Theorem (Armond and Dasbach) Let K 1 and K 2 be two alternating links with alternating diagrams D 1 and D 2 such that the reduced A-checkerboard (respectively B-checkerboard) graphs of D 1 and D 2 coincide. Then the tails(respectively heads) of the colored Jones polynomial of K 1 and K 2 are identical.

45 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Figure: The Knot 6 2

46 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Figure: The Knot 6 2 with a checkerboard coloring

47 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Figure: The Knot 6 2 with one of its associated graph

48 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Figure: The Knot 6 2 with the other associated graph

49 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Knot Knot Diagram White Checkerboard Graph Checkerboard Graph Black Tail Head 3_1 1 h 3 4_1 h 3 h 3 5_1 h 5 1 5_2 h 3 * h 4 6_2 h 3 * h 3 h 4 7_4 h 3 (h 4 ) 2 7_7 (h 3 ) 2 (h 3 ) 3 8_5 h 3??? h b (q) = ( 1) n q bn(n+1)/2 n n Z h b (q) = n Z ɛ(n)q bn(n+1)/2 n

50 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence

51 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence For the figure 8 knot Φ 0 = (1 q n ) = n=1 Φ 0. ( 1) k q k 2 (3k 1). k= Φ N = N = N =

52 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Φ N = N = N = Now, since we know all of Φ 0, we can subtract it from the shifted colored Jones polynomials. Now are coefficients are: Φ N = N = N =

53 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Φ N = N = N = Shifting these sequences back so that they start with a non-zero term, we can see that they again stabilize. The sequence they stabilize to is Φ 1. Φ N = N = N =

54 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Definition ([7]) A sequence f n (q) Z[[q]] is k-stable if there exist Φ j (q) Z((q)) for j = 0,..., k such that ( lim n q k(n+1) f n (q) k ) Φ j (q)q j(n+1) = 0 j=0 A sequence is stable if it is k stable for all k. Let J K,n be the unnormalized colored Jones polynomial for the knot K colored with the n + 1-dimensional representation. (Different from the original convention) Let Ĵ K,n be J K,n divided by its lowest monomial.

55 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Let J K,n be the unnormalized colored Jones polynomial for the knot K colored with the n + 1-dimensional representation. (Different from the original convention) Let Ĵ K,n be J K,n divided by its lowest monomial. Theorem ([7]) For every alternating link K, the sequence (Ĵ K,n (q)) is stable and its associated k-limit Φ K,k (q) can be effectively computed from any reduced, alternating diagram D of K.

56 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence m 1 m 2 m 3 Figure: A trefoil knot with its checkerboard graph.

57 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence Theorem (W.) The tailneck of knots with reduce to the three cycle is: n=1 (1 qn ), i.e. the pentagonal numbers sequence, if all m i = 1 (The only knot satisfying this is the trefoil). n=1 (1 qn n=1 ) + (1 qn ) 1 q, i.e. the pentagonal numbers plus the partial sum of the pentagonal numbers, if two m i = 1 and one is 2 or more. n=1 (1 qn n=1 ) + 2 (1 qn ) 1 q, i.e. the pentagonal numbers plus the 2 times the partial sum of the pentagonal numbers, if one m i = 1 and two are 2 or more. n=1 (1 qn n=1 ) + 3 (1 qn ) 1 q, i.e. the pentagonal numbers plus the 3 times the partial sum of the pentagonal numbers, if all m i 2.

58 Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence 1 Can we use the stablized sequences to help us find the middle coefficients? 2 Are there other ways to calculate the colored Jones polynomial that help us understand the coefficients. 3 What can we say about the patterns in other knots like non-alternating knots?

59 Any Questions?

60 Selected References Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence [1] C. Armond. The head and tail conjecture for alternating knots. ArXiv e-prints, December [2] C. Armond and O. T. Dasbach. Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial. ArXiv e-prints, June [3] Dror Bar-Natan, Scott Morrison, and et al. The Knot Atlas. [4] A. Champanerkar and I. Kofman. On the tail of Jones polynomials of closed braids with a full twist. ArXiv e-prints, April [5] O. Dasbach and X.-S. Lin. On the head and the tail of the colored jones polynomial. Compos. Math., 5: , [6] O. Dasbach and X.-S. Lin. A volumish theorem for the jones polynomial of alternating knots. Pacific J. Math., 2: , [7] S. Garoufalidis and T. T. Q. Le. Nahm sums, stability and the colored Jones polynomial. ArXiv e-prints, December [8] Stavros Garoufalidis and Thang T Q Le. Asymptotics of the colored jones function of a knot. Geom. and Topo., 15: , [9] W. B. R. Lickorish. An Introduction to Knot Theory. Springer, [10] G. Masbaum. Skein-theoretical derivation of some formulas of habiro. Algebr. Geom. Topol., 3: , [11] H. Murakami. An Introduction to the Volume Conjecture. ArXiv e-prints, January [12] Dylan Thurston. Hyperbolic volume and the jones polynomial: A conjecture. dpt/speaking/hypvol.pdf.