Advancement to Candidacy. Patterns in the Coefficients of the Colored Jones Polynomial. Katie Walsh Advisor: Justin Roberts

Transcription

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

2 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 1 Basics of Knot Theory Introduction Knots and Knot Invariants Jones Polynomial and Generalizations 2 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 3 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot 4 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem

3 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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?

4 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations Definition ([8]) 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.

5 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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.

6 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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:

7 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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

8 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations Reidemeister 2: Reidemeister 3:

9 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations Reidemeister 2: Reidemeister 3: Reidemeister 1:

10 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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) = # #

11 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations We can also view the Jones Polynomial as a specific case of the Reshetikhin-Turaev invariant of the knot. Consider the Lie algebra sl 2 of traceless two-by-two complex matrices. It has universal enveloping algebra U(sl 2 ), which has quantum deformation U q (sl 2 ). U q (sl 2 ) is an algebra over the ring Q(q) of rational functions in the determinant q. Given a knot diagram, we can color the diagram with an 2 dimensional representation of U q (sl 2 ) V.

12 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 1 (V V ) V V V V V V V V Id R Id (V V ) 1

13 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 1 (V V ) (V V ) V V V V V V V V V V V V V V V (V V) V (V V ) 1 Id Id R Id Id R Id R 1 Id Id R 1 Id Id Id Id This produces a map from 1 to itself and thus is just multiplication by an element in Q(q).

14 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations Knot Knot Diagram Jones Polynomial Trefoil (3 1 ) q +q 3 q 4 Figure Eight (4 1 ) q 2 q 1 +1 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

15 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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 1 +1 q +q 2 Mirror Image (4 1 ) q 2 q 1 +1 q 1 +q 2 Table: Jones Polynomials of Knot and their Mirror Images

16 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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

17 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations We can generalize the construction of the Jones Polynomial by letting V be a N dimensional representation of U q (sl 2 ). 1 (V V ) (V V ) V V V V V V V V V V V V V V V (V V) V (V V ) 1 Id Id R Id Id R Id R 1 Id Id R 1 Id Id Id Id This gives us the N dimensional colored Jones polynomial.

18 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations 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

19 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations We can think of the N dimensional colored Jones polynomials as a linear combination of the original Jones polynomial on cablings of the knots, in the same way that the N dimensional representation can be expressed as a linear combination of V2 k,k n 1. We can express this linear combination recursively as: g 1 = 1 g 2 = z g 3 = zg i 1 g i 2. For example, g 3 = z 2 1 so the 3 dimensional colored Jones polynomial is the Jones Polynomial of the two cabled figure 8 minus 1.

20 Introduction Knots and Knot Invariants The Jones Polynomial and Generalizations J 3,41 = V( ) 1

21 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial We will now define the n th Temperley-Lieb Algebra, TL n and the Jones-Wenzl idempotent f (n) TL n Figure: Start with an oriented disk (D 2 ) with 2n marked points in its boundary.

22 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial Figure: We draw arcs starting and ending at all of the marked points and cycles inside the surface. We have to mark at each crossing which strand is the over crossing and which is the under crossing.

23 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial Figure: Two diagrams are the same is there is a homeomorphism of D 2 that is isotopic to the identity and keeps the boundary points fixed that maps ones to the other.

24 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial Definition The linear skein S(D 2,2n) of (D 2,2n) is a vector space of formal linear sums over C of link diagrams in (D 2,2n) quotiented by the relations below. D = ( A 2 A 2 )D = A +A 1

25 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The product of diagrams is by juxtaposition.

26 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The product of diagrams is by juxtaposition.

27 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The product of diagrams is by juxtaposition.

28 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The product of diagrams is by juxtaposition.

29 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial This product extends to a well-defined bilinear map that turns S(D 2,2n) into an algebra, the n th Temperley-Lieb algebra TL n.

30 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial This product extends to a well-defined bilinear map that turns S(D 2,2n) into an algebra, the n th Temperley-Lieb algebra TL n. The algebra is generated by the elements 1,e 1,e 2,...,e n 1 shown below. 1 = n e i = n -i -1 i -1 Here, a strand with an n above it represents n parallel strands in the diagram.

31 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial f (n) = n The Jones-Wenzl idempotent f (n) TL n is the unique element such that: (i) f (n) e i = 0 = e i f (n) for 1 i n 1 (kills backtracks) (ii)(f (n) 1) belongs to algebra generated by {e 1...e n 1 } (iii) f (n) f (n) = f (n) (iv) n = ( 1)n (A 2(n+1) A 2(n+1) ) (A 2 A 2 ) n = n S(R 2 )

32 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial Wenzl proved the following recurrence relationship satisfied by the idempotents: f (0) = the empty diagram f (1) = 1 1 n+1 = n - n 1 n 1 1 n n-1 n

33 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 1 2 =

34 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 1 2 = = 1 - A 2 A 2 A 4 A 4 1 1

35 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 3 =

36 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The (N +1) st colored Jones polynomial of a knot K is the Jones polynomial of K decorated with the f (N), the Jones-Wenzl idempotent in TL n.

37 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial The (N +1) st colored Jones polynomial of a knot K is the Jones polynomial of K decorated with the f (N), the Jones-Wenzl idempotent in TL n. We will consider the normalized colored Jones Polynomial J N,K (q), which is normalized such that J N,unknot (q) = 1. J N,K (q) = J N,K(q) n

38 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial There are various formulas to calculate the colored Jones Polynomial for different classes of knots including the figure eight knot, twist knots with p full twists K p pictured on the left below and (1,2p 1,l 1) pretzel knots, K p,l pictured on the right. p full twists 2p-1 total half twists l-1 total half twists

39 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 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}.

40 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 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 i2 +2i a a 1

41 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 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

42 The Temperley-Lieb Algebra Other Definitions of the Colored Jones Polynomial 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)

43 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot 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 14 +3q 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}

44 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot {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

45 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 20 th Colored Jones Polynomial for the Figure Eight Knot

46 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 50 th Colored Jones Polynomial for the Figure Eight Knot

47 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 95 th Colored Jones Polynomial for the Figure Eight Knot

48 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 10 th Colored Jones Polynomial for the 2 Twist Knot (5 2 )

49 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 20 th Colored Jones Polynomial for the 2 Twist Knot (5 2 )

50 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 30 th Colored Jones Polynomial for the 2 Twist Knot (5 2 )

51 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 20 th Colored Jones Polynomial for the 3 Twist Knot (7 2 )

52 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 15 th Colored Jones Polynomial for the (1,3,5) Pretzel Knot (9 5 )

53 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 95 th Colored Jones Polynomial for the Figure Eight Knot

54 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Figure: Coefficients of the 95 th Colored Jones Polynomial for the Figure Eight Knot Divided by Sin

55 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot 1 In the middle, the coefficients are periodic with period N, the number of colors. 2 There is a sine wave like oscillation on the first and last quarter of the coefficients. 3 We can see that the sine oscillation persists throughout the entire polynomial. It s amplitude seems to grow with the shape of a normal distribution on the exterior and is very steady in the middle.

56 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Constant Coefficient of the Colored Jones Polynomial of the Figure 8 Knot Constant Coefficient Number of Colors

57 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Normalized Growth Rate of the Constant Term ln(constant Coef)*2 /N Number of Colors

58 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot 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.

59 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Knot Knot Diagram Volume Trefoil (3 1 ) Not Hyperbolic Figure Eight (4 1 ) Not Hyperbolic Table: Hyperbolic Volumes of Different Knots

60 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot 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 [10]).

61 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot If we let m K (N) be the maximum coefficient of the Colored Jones Polynomial of a knot K, then J N,K (e2πi/n ) M m K (N) Mm K (N), n=0 where M is the number of terms in the colored Jones polynomial. For alternating knots, M = N(N 1)c/2 while for all knots the growth rate of the breadth is at most quadratic, so M a 2 N 2 +a 1 N +a 0.

62 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Thus, log J k (N)(e 2πi/N ) log(a 2 N 2 +a 1 N +a 0 )m K (N) lim lim N N N N log(a 2 N 2 +a 1 N +a 0 ) = lim N = lim N N logm K (N). N + logm K(N) N So for knots where the Hyperbolic Volume Conjecture holds, this would say that. vol(s 3 2πlogm K (N) \K) lim N N

63 Visual Representations of the Patterns The Conjectures Hyperbolic Volume of a Knot Normalized Growth Rate of the Constant Term ln(constant Coef)*2 /N Number of Colors

64 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem Not much other work had been done studying the middle coefficients on the colored Jones polynomial but there is current work being done studying patterns and stabilization in the other coefficients. This work has also been related to the Hyperbolic Volume Conjecture. There is also work being done by Garoufalidis studying the asymptotic behavior and recurrence relations of the colored Jones polynomial for a knot.

65 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem Definition The head of the Colored Jones Polynomial of a knot K - if it exists - is a polynomial whose first N terms (highest powers of q) have the same coefficients as the first N terms of J N,K. Definition The tail of the Colored Jones Polynomial of a knot K - if it exists - is a polynomial whose last N terms (lowest powers of q) have the same coefficients as the first N terms of J N,K.

66 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem 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 16 +3q 15 q 14 q q 30 q 29 q 28 +0q 27 +0q 26 +q 25 +2q 24 +0q q 42 q 41 q 40 +0q 39 +0q 38 +q 37 +0q 36 +3q q 56 q 55 q 54 +0q 53 +0q 52 +q 51 +0q 50 +q 49 +

67 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem The Head of the Colored Jones Polynomial of 4 1 J N,4 1 (q) = = = N 1 n=0 k=1 N 1 n=0 k=1 N 1 n n=0 k=1 n {N k}{n +k} n (a N k a N+k )(a N+k a N k ) q N q k q k +q N The max degree of each summand is Nn so decreasing the n by 1 changes the max degree by N thus only n = N 1 contributes to the head and tail.

68 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem J N,4 1 (q) = reindex:k = N k N 1 N 1 HT = n=0 k=1 k=1 N 1 HT = k=1 N 1 = q N N 1 HT = k =1 n q N q k q k +q N q N q k q k +q N q N q k k=1 1 q k N (1 q k )

69 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem J N,4 1 (q) HT = N 1 k =1 (1 q k ) Theorem (Euler s Pentagonal Number Theorem) (1 x n ) = ( 1) k x k(3k 1)/2 n=1 k= = 1 x x 2 +x 5 +x 7 x 12 x 15 + A similar arguments shows that the head of twists knots is the same polynomial.

70 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem In 2006, Dasbach and Lin conjectured that the head and tail exist for alternating knots. In 2011, Armond proved this for alternating links and for adequate links, using skein theoretical techniques. The head and tail do not exist for all knots, however. Armond and Dasbach showed that the head and tail does not exist for the (4,3) torus knot.

71 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem The tails are known for the following knots: All knots up to 7 crossing (2,p) torus knots twist knots generalized twist knots 2-bridge knots

72 Head and Tail of the Colored Jones Polynomial Volume-ish Theorem 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.

73 Conjectures and Questions Head and Tail of the Colored Jones Polynomial Volume-ish Theorem 1 The coefficients of J N,K are periodic with period N, especially for the middle terms as N gets large. 2 There is a sinusoidal oscillation in the first and last quarter of the coefficients (but after the head and the tail). 3 The growth rate of the max coefficient is exponential and related to the hyperbolic volume of the knot. 4 How much does head and tail of the colored Jones polynomial tell us about the knot? 5 How much do the middle coefficients tell us?

74 Thanks for Coming!!

75 Selected References Head and Tail of the Colored Jones Polynomial Volume-ish Theorem [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] Stavros Garoufalidis and Thang T Q Le. Asymptotics of the colored jones function of a knot. Geom. and Topo., 15: , [8] W. B. R. Lickorish. An Introduction to Knot Theory. Springer, [9] G. Masbaum. Skein-theoretical derivation of some formulas of habiro. Algebr. Geom. Topol., 3: , [10] H. Murakami. An Introduction to the Volume Conjecture. ArXiv e-prints, January [11] Dylan Thurston. Hyperbolic volume and the jones polynomial: A conjecture. dpt/speaking/hypvol.pdf.