Patterns and Stability in the Coefficients of the Colored Jones

Home , Hyperbolic volume, The Knot Atlas

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-1 ´ 1012 Katie Walsh Advisor: Justin Roberts

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-3 ´ 1012 Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

1 Motivation The Middle Coeﬃcients of the Colored Jones Polynomial Jones Polynomial

2 Knots and the Jones Polynomial Introduction

3 The Colored Jones Polynomial Deﬁnitions Hyperbolic Volume Conjecture

4 Patterns in the Colored Jones Coeﬃcients Head and Tail of the Colored Jones Polynomial Stability of the Colored Jones Sequence

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

The 5th colored Jones Polynomial for ﬁgure 8 knot is: 1 1 1 3 1 1 1 1 5 1 2 2 1 − − + − − − − + − − − − q20 q19 q18 q15 q14 q13 q12 q11 q10 q9 q8 q7 q6 6 1 2 2 1 + − − − − +7−q−2q2−2q3−q4+6q5−q6−2q7−2q8−q9+5q10 q5 q4 q3 q2 q −q11 − q12 − q13 − q14 + 3q15 − q18 − q19 + q20 This has coeﬃcients:

{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}

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

{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:

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Figure: Coeﬃcients of the 5th Colored Jones Polynomial for the Figure Eight Knot

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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Figure: Coeﬃcients of the 95th Colored Jones Polynomial for the Figure Eight Knot Divided by “Sin”

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Constant Coefficient of the Colored Jones Polynomial of the Figure 8Knot 5000 4500 4000 3500 3000 Coefficient

2500 2000

Constant 1500 1000 500 0 0 5 10 15 20 25 30 35 Number of Colors

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Normalized Growth Rate of the Constant Term 3

2.5 /N

ʋ 2 Coef)*2 1.5

1 ln(Constant

0.5

0 0 5 10 15 20 25 30 35 40 45 50 Number of Colors

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Volume

Trefoil (31) Not Hyperbolic

Figure Eight (41) 2.0298832132

51 Not Hyperbolic

10132 4.05686 Table: Hyperbolic Volumes of Diﬀerent Knots

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

1 In the middle, the coefficients of JK,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.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial The Middle Coeﬃcients of the Colored Jones Polynomial The Colored Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Deﬁnition A knot is an embedding f :S1 → S3.

2 A knot is usually represented through projection into R 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?

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Deﬁnition ([9]) Two knots are equivalent if there is an orientation preserving piecewise linear homeomorphism h : S3 → S3 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.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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: ←→

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Deﬁnition The Kauﬀman Bracket is an invariant of framed knots. It is characterized by the skein relation below.

= 1

D t = (−A2 − A−2) D

= A + A−1

Katie Walsh WimSoCal Reidemeister 1: =

Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Reidemeister 2: ←→

Reidemeister 3: ←→

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Reidemeister 2: ←→

Reidemeister 3: ←→

Reidemeister 1: =

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

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) hDi q1/2=A−2 where w(D) is the writhe of the diagram.

w(D) = # −#

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Jones Polynomial

3 4 Trefoil (31) q + q − q

−2 −1 2 Figure Eight (41) q − q + 1 − q + q

2 4 5 6 7 51 q + q − q + q − q

−2 −4 −5 −6 −7 10132 q + q − q + q − q Table: Jones Polynomials of Diﬀerent Knots

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Jones Polynomial

3 4 Trefoil (31) q + q − q

−1 −3 −4 Mirror Image(31) q + q − q

−2 −1 2 Figure Eight (41) q − q + 1 − q + q

2 1 −1 −2 Mirror Image (41) q − q + 1 − q + q Table: Jones Polynomials of Knot and their Mirror Images

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Introduction The Colored Jones Polynomial The Jones Polynomial Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Jones Polynomial

2 4 5 6 7 51 q + q − q + q − q

−2 −4 −5 −6 −7 51 q + q − q + q − q

−2 −4 −5 −6 −7 10132 q + q − q + q − q Table: Jones Polynomials of Diﬀerent Knots

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

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”.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Colored Jones Polynomial (2+1 dim rep)

q−2 + q−5 − q−7 + q−8 − q−9 Trefoil (31) −q−10 + q−11

q6 − q5 − q4 + 2q3 − q2 − q+ Figure Eight (41) 3 − q−1 + −q−2 + 2q −3 − q−4 − q−5 + q−6

q−4 + q−7 − q−9 + q−10 − q−12+ 51 q−13 − 2q−15 + q−16 − q−18 + q−19

−q + 1 + 2q−1 − 3q−2 + q−3+ 3q−4 − 4q−5 + 2q−6 + 2q−7 − 3q−8+ 10132 2q−9 + q−10 − 3q−11 + 2q−12 − 2q−14+ 2q−15 − q−16 − q−17 + 2q−18 − q−19− q−20 + q−21 Table: Colored Jones Polynomials of Diﬀerent Knots

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients 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:

g1 = 1

g2 = z

gi = zgi−1 − gi−2. 2 For example, g3 = z − 1 so,

J3,41 = V ( ) − 1

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

We have formulas for the ﬁgure eight knot, twist knots, Kp and (1, 2p − 1, r − 1) pretzel knots, Kp,r .

2p-1 r-1 p full twists

The twist knot Kp The (1, 2p − 1, r − 1) pretzel knot

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Theorem (Habiro and Le)

N−1 X {N − n}{N − 1 + 1}···{N + n} J0 (a2) = N,41 {N} n=0 where {n} = an − a−n and {n}! = {n}{n − 1}···{1}.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Theorem (Habiro and Le) For a twist knots with p twists,

N−1 X {N − n}{N − 1 + 1}···{N + n} J0 (a2) = f N,Kp Kp,n {N} n=0 where

n 1 X [n]! f = an(n+3)/2 (−1)k µp [2k+1] Kp,n (a − a−1)n 2k [n + k + 1]![n − k]! k=0 As standard, an − a−n q = a2, a = A2, {n} = an − a−n, [n] = a − a−1 and i i 2+2i µi = (−1) A Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Theorem (W.)

A pretzel knot of the form Kp,l = P(1, 2p − 1, l − 1) has the colored Jones polynomial

l/2 n h i (−1)k(l+1)[2k+1]µ PN−1 (−1) N+n c0 {2n+1}!{n}! Pn 2k n=0 (a−a−1)2n N−n−1 n,p {1} k=0 [n+k+1]![n−k]! J0 (a2) = . N,Kp,l [N]

When l is even this reduces to

PN−1 n N+n 0 {2n+1}! 0 n=0 (−1) N−n−1 cn,p cn,l/2 J0 (a2) = {1} . n,Kp,l [N]

Here n 1 X [n]! c0 = (−1)k µp [2k + 1] . n,p (a − a−1)n 2k [n + k + 1]![n − k]! k=0

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Knot Twists Pretzel Notation (p,l)

31 1 (1,3,0) or (1,1,1) (2,1) or (1,2) 41 (1,1,2) (1,3) 51 (1,5,0) (3,1) 52 2 (1,3,1) or (1,1,3) (2,2) or (1,4) 61 (1,1,4) (1,5) 62 (1,3,2) (2,3) 71 (1,7,0) (4,1) 72 3 (1,1,5) or (1,5,1) (1,6) or (3,2) 74 (1,3,3) (2,4) 81 (1,1,6) (1,7) 82 (1,5,2) (3,3) 84 (1,3,4) (2,5)

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Deﬁnition The hyperbolic volume of a hyperbolic knot K is the volume of the unique hyperbolic metric on the knot complement (S3 \ 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.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Knot Knot Diagram Volume

Trefoil (31) Not Hyperbolic

Figure Eight (41) 2.0298832132

51 Not Hyperbolic

10132 4.05686 Table: Hyperbolic Volumes of Diﬀerent Knots

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

Conjecture (Kashaev, Murakami, Marakami) The Hyperbolic Volume Conjecture states that:

log |J0 (e2πi/N )| vol(S3 \ K) = 2π lim N,K 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]).

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

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 n m JK,2(q) = anq + ··· + amq be the Jones polynomial of K. Then

3 2v0(max(|am−1|, |an+1|) − 1) ≤ Vol(S − K)

3 Vol(S − K) ≤ 10v0(|an+1| + |am−1| − 1).

Here, v0 ≈ 1.0149416 is the volume of and ideal regular hyperbolic tetrahedron.

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Deﬁnitions The Colored Jones Polynomial Hyperbolic Volume Conjecture Patterns in the Colored Jones Coeﬃcients

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

Katie Walsh WimSoCal Motivation Knots and the Jones Polynomial Head and Tail of the Colored Jones Polynomial The Colored Jones Polynomial Stability of the Colored Jones Sequence Patterns in the Colored Jones Coeﬃcients

N Highest Terms of the Colored Jones Polynomial of 41 2 q2−q + 1 − q−1 + q−2 3 q6−q5−q4 + 2q3 − q2 − q + 3 − q−1 − q−2 + ··· 4 q12−q11−q10+0q9 + 2q8 − 2q6 + 3q4 − 3q2 + ··· 5 q20−q19−q18+0q17+0q16 + 3q15 − q14 − q13 + ··· 6 q30−q29−q28+0q27+0q26+q25 + 2q24 + 0q23 + ··· 7 q42−q41−q40+0q39+0q38+q37+0q36 + 3q35 + ··· 8 q56−q55−q54+0q53+0q52+q51+0q50+q49 + ···

The Head of the Colored Jones Polynomial of 41

N−1 n X Y J0 (q) = {N − k}{N + k} N,41 n=0 k=1 N−1 HT Y 0 J0 (q) = (1 − q−k ) N,41 k0=1

Theorem (Euler’s Pentagonal Number Theorem)

∞ ∞ Y X (1 − xn) = (−1)k xk(3k−1)/2 n=1 k=−∞ = 1 − x − x2 + x5 + x7 − x12 − x15 + ···

Theorem (Armond and Dasbach)

Let K1 and K2 be two alternating links with alternating diagrams D1 and D2 such that the reduced A-checkerboard (respectively B-checkerboard) graphs of D1 and D2 coincide. Then the tails(respectively heads) of the colored Jones polynomial of K1 and K2 are identical.

Figure: The Knot 62

Figure: The Knot 62 with a checkerboard coloring

Figure: The Knot 62 with one of its associated graph

Figure: The Knot 62 with the other associated graph

Knot Diagram “White” “Black” Knot Checkerboard Checkerboard Graph Tail Head Graph

3_1 1 h3

4_1 h3 h3

5_1 h5 1

* 5_2 h3 h4

* 6_2 h3 h3h4

2 7_4 h3 (h4)

2 3 7_7 (h3) (h3)

8_5 h3 ???

X n bn(n+1)/2−n ∗ X bn(n+1)/2−n hb(q) = (−1) q hb(q) = (n)q n∈Z n∈Z

For the ﬁgure 8 knot

∞ ∞ Y n X k k (3k−1) Φ0 = (1 − q ) = (−1) q 2 . n=1 k=−∞

Φ0.

Φ0 1 -1 -1 0 0 1 0 1 0 0 0 0 -1 ··· N = 3 1 -1 -1 0 2 0 -2 0 3 0 -3 0 3 ··· N = 4 1 -1 -1 0 0 3 -1 -1 -1 -1 5 -1 -2 ··· N = 5 1 -1 -1 0 0 1 2 0 -2 -1 -1 1 3 ···

Now, since we know all of Φ0, we can subtract it from the shifted colored Jones polynomials. Now are coeﬃcients are:

Φ0 1 -1 -1 0 0 1 0 1 0 0 0 0 -1 ··· N = 3 0 0 0 0 2 -1 -2 -1 3 0 -3 0 4 ··· N = 4 0 0 0 0 0 2 -1 -2 -1 -1 5 -1 -3 ··· N = 5 0 0 0 0 0 0 2 -1 -2 -1 -1 1 4 ···

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 Φ1.

Φ1 2 -1 -2 -1 -1 1 ··· N = 3 2 -1 -2 -1 3 0 -3 0 4 0 -3 1 ··· N = 4 2 -1 -2 -1 -1 5 -1 -3 -2 -1 7 ··· N = 5 2 -1 -2 -1 -1 1 4 1 -2 -2 ···

Deﬁnition ([7])

A sequence fn(q) ∈ Z[[q]] is k-stable if there exist Φj (q) ∈ Z((q)) for j = 0,..., k such that

k −k(n+1) X j(n+1) lim q fn(q) − Φj (q)q = 0 n→∞ j=0

A sequence is stable if it is k−stable for all k.

Let JK,n be the unnormalized colored Jones polynomial for the knot K colored with the n + 1-dimensional representation. (Diﬀerent from the original convention) Let ˆJK,n be JK,n divided by its lowest monomial.

For every alternating link K, the sequence (JˆK,n(q)) is stable and its associated k-limit ΦK,k (q) can be eﬀectively computed from any reduced, alternating diagram D of K.

Figure: A trefoil knot with its checkerboard graph.

Theorem (W.) The tailneck of knots with reduce to the three cycle is: Q∞ n n=1(1 − q ), i.e. the pentagonal numbers sequence, if all mi = 1 (The only knot satisfying this is the trefoil). Q∞ n Q∞ n n=1(1−q ) n=1(1 − q ) + 1−q , i.e. the pentagonal numbers plus the partial sum of the pentagonal numbers, if two mi = 1 and one is 2 or more. Q∞ n Q∞ n n=1(1−q ) n=1(1 − q ) + 2 1−q , i.e. the pentagonal numbers plus the 2 times the partial sum of the pentagonal numbers, if one mi = 1 and two are 2 or more. Q∞ n Q∞ n n=1(1−q ) n=1(1 − q ) + 3 1−q , i.e. the pentagonal numbers plus the 3 times the partial sum of the pentagonal numbers, if all mi ≥ 2.

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?

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-3 ´ 1012 Motivation Knots and the Jones Polynomial Head and Tail of the Colored Jones Polynomial The Colored Jones Polynomial Stability of the Colored Jones Sequence Patterns in the Colored Jones Coeﬃcients Selected References

[1] C. Armond. The head and tail conjecture for alternating knots. ArXiv e-prints, December 2011. [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 2011. [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 2010. [5] O. Dasbach and X.-S. Lin. On the head and the tail of the colored jones polynomial. Compos. Math., 5:1332–1342, 2006. [6] O. Dasbach and X.-S. Lin. A volumish theorem for the jones polynomial of alternating knots. Paciﬁc J. Math., 2:279–291, 2007. [7] S. Garoufalidis and T. T. Q. Le. Nahm sums, stability and the colored Jones polynomial. ArXiv e-prints, December 2011. [8] Stavros Garoufalidis and Thang T Q Le. Asymptotics of the colored jones function of a knot. Geom. and Topo., 15:2135–2180, 2011. [9] W. B. R. Lickorish. An Introduction to Knot Theory. Springer, 1997. [10] G. Masbaum. Skein-theoretical derivation of some formulas of habiro. Algebr. Geom. Topol., 3:537–556, 2003. [11] H. Murakami. An Introduction to the Volume Conjecture. ArXiv e-prints, January 2010. [12] Dylan Thurston. Hyperbolic volume and the jones polynomial: A conjecture. http://www.math.columbia.edu/ dpt/speaking/hypvol.pdf.

Katie Walsh WimSoCal