3 ´ 1012 2 ´ 1012 1 ´ 1012 Patterns and Stability in the Coefficients of the Colored Jones Polynomial 5000 10 000 15 000 -1 ´ 1012 Katie Walsh Advisor: Justin Roberts -2 ´ 1012 -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 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 WimSoCal5 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: f1; −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; 1g 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 f1; −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; 1g We can plot these: 6 4 2 10 20 30 40 -2 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 200 100 100 200 300 400 500 600 700 -100 -200 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 2 ´ 106 1 ´ 106 1000 2000 3000 4000 5000 -1 ´ 106 -2 ´ 106 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 3 ´ 1012 2 ´ 1012 1 ´ 1012 5000 10 000 15 000 -1 ´ 1012 -2 ´ 1012 -3 ´ 1012 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 20 10 5000 10 000 15 000 -10 -20 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.