My research is in the area of topology, specifically in knot theory. The bulk of my research has been on the patterns in the coefficients of the colored Jones polynomial. The colored Jones polynomial,j K,N (q), is a knot invariant of a knot K that assigns to each knot a sequence of Laurent polynomials indexed by N 2. When N = 2, we get the Jones polynomial. We usually think of the N-colored Jones polynomial as either the Jones polynomial of a linear combination of i-cablings of the knot for 0 i N 1 or as the evaluation in the Temperley-Lieb algebra of the knot diagram decorated with the N 1 st Jones- Wenzl idempotent. One of the main open questions about the colored Jones polynomial is how to relate it to the geometry of the knot. One such relation is the following hyperbolic volume conjecture. Conjecture 1 ([Mur10], Kashaev-Murakami-Murakami). For any hyperbolic knot K, log J K,N (e 2πi/N ) 2π lim N N = vol(s 3 \K) where J K,N (e 2πi/N ) is the normalized Colored Jones Polynomial of a knot K evaluated at an N th root of unity and vol(s 3 \K) is the volume of the unique complete hyperbolic Riemannian metric on the knot complement. The hyperbolic volume conjecture has been proved for various knots and knot families include the figure 8 knot and torus knots but is still open for many other knots and links. In [Das], Dasbach and Lin related the first and last two coefficients of the original Jones polynomial to the the volume of the knot in the following way: Theorem 0.1 (Dasback, Lin). Volume-ish Theorem: For an alternating, prime, nontorus knot K let J K,2 (q) = a n q n + + a m q m be the Jones polynomial of K. Then 2v 8 (max( a m 1, a n+1 ) 1) Vol(S 3 K) 10v 3 ( a n+1 + a m 1 1). Here, v 3 1.0149416 is the volume of an ideal regular hyperbolic tetrahedron and v 8 3.66386 is the volume of an ideal regular hyperbolic octahedron. They also proved that the first two and last two coefficients of the Jones Polynomial where also the first and last two coefficients of the N-colored Jones polynomial for all N and conjectured that for some knots the first and last N coefficents of the N-colored Jones polynomial are the same, up to sign, as the first N coefficients of the k-colored Jones polynomial for all k > N. We will discuss this further in section 3. These types of theorems led us to begin looking more deeply in to what the coefficients of the colored Jones polynomial can tell us about the knot. 1
Katherine Walsh Research Statement 1 October 2013 Patterns in the Coefficients of the Colored Jones Polynomial When studying the coefficients of the colored Jones polynomial, I first looked at patterns in the entire set of coefficients. To be able to visualize these patterns, I used a formula initially proved by Habiro and reproved by Masbaum in [Mas03] to calculate the colored Jones polynomial of the figure 8 knot and twist knots and then plotted the coefficients of these polynomials. The plot of the coefficients for the 95th colored Jones polynomial of the figure 8 knot and the 30th colored Jones polynomial of the knot 52 are below. (The plot has the degree of the term on the x axis and the coefficient on the y axis. Degrees were shifted by multiplying by q M for some M so that all the degrees were positive.) 3 1012 2 1012 150 000 1 1012 100 000 5000 10 000 50 000 15 000-1 1012 500 1000 1500 2000-50 000-2 1012-100 000-3 1012-150 000 (a) Coefficients of the 95th Colored Jones Polynomial for the Figure Eight Knot (b) Coefficients of the 30th Colored Jones Polynomial for the Knot 52 Figure 1 This led me to the following conjectures about the basic shape of the plot of the coefficients of the N th colored Jones polynomial. 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. I also looked at the growth rate of the maximum coefficients of each colored Jones polynomial of a knot. The maximum coefficients of the polynomials seemed to grow exponentially at a rate related to the hyperbolic volume of the knot. 2
2 A formula for the Colored Jones Polynomial of a (1, r 1, 2p 1 pretzel knots Much of my research has been centered on trying to gain insight on where these patterns come from. This first led me to use the techniques from [Mas03] to find a formula for the colored Jones polynomial of pretzel knots of the form (1, r 1, 2p 1) in order to have a larger class of knots for which I could easily calculate the colored Jones polynomial for large values of N. c 1 c 2 c 3 c n Figure 2: A (c 1, c 2,..., c n ) Pretzel Knot. A box with a c i represents c i half twists. A pretzel knot or link is usually described by P(c 1, c 2,... c n ) where each c i is an integer representing the number of half twists within that section of the knot. These twisted parts are drawn vertically. Positive c i correspond with positive half twists, while negative c i correspond with negative half twists. See Figure 2. We consider pretzel knots of the form P (1, 2p 1, r 1). Many of the knots with a small number of crossings can be expressed as a pretzel knot of this form. There are only 4 knots with up to 9 crossings tht can be expressed as twsit knots but 15 that can be written in this way. Following the techniques of Masbaum in [Mas03] we can find the following formula for the colored Jones polynomial. Theorem 2.1. A pretzel knot of the form K p,r = P (1, 2p 1, r 1) has the colored Jones polynomial Here J N (K p,r, a 2 ) = N 1 = N 1 n=0 ( 1)n[ N+n N n 1 c n,p = [ N+n n=0 c n,p N 1 n 1 (a a 1 ) n where µ i = ( 1) i A i2 +2i and, ] c n,p {2n+1}!{n}! {1} ] µ n n k=0 δ(2k; n, n)r 2k n,n,2k ([k]!) 2 [2k]! {2n+1}! {n}!{1} 1 n (a a 1 ) 2n k=0 ( 1)k(r+1) [2k+1] [n+k+1]![n k]! µr/2 2k. n ( 1) k µ p [n]! 2k [2k + 1] [n + k + 1]![n k]!, k=0 {n} = a n a n, [n] = an a n a a 1 3
[ ] n := k [n]! [k]![n k]!. Corollary 2.2. When r is even this reduces to J n (K p,r, a 2 ) = N 1 n=0 [ ] N + n ( 1) n N n 1 c n,p {2n + 1}! c {1} n,r/2. Corollary 2.3. When r is odd this reduces to N 1 [ ] N + n {2n + 1}!{n}! J n (K p,r, a 2 ) = ( 1) n µ 4p n c n,p N 1 n (a a 1 ) 2n {1} n=0 n k=0 µ 2k r 2 [2k + 1] [n + k + 1]![n k]! The formula for the case where r is even was independently proven by Garoufalidis and Koutschan in [GK12]. Using this formula, we are able to more quickly calculate the colored Jones polynomial for many knots with up to 9 crossings. 3 The Head and Tail and Higher Order Stability Given a sequence of Laurent polynomials, we say the head of this polynomial exists if the first N coefficients (of the highest order terms) of the N th polynomial in the sequence are the same as the first N coefficients of the k th polynomial for all k N. The tail of the sequence of polynomials, if it exists, is the stabilized sequences of the coefficients of the lowest terms. In [DL06, AD11, Arm11], Dasbach and Armond proved that the head and tail of the colored Jones exist for alternating and adequate knots and depend on the reduced checkerboard graphs of the knot diagrams. For example, for the figure 8 knot, we know the that first coefficients stabilize to the pentagonal number sequence. By this, I mean that for the figure 8 knot, Φ 0 = (1 q n ) = n=1 k= ( 1) k q k 2 (3k 1). In the table below, I have listed out the first 16 coefficients of the N-colored Jones polynomial for the figure 8 knot for N = 3, 4 and 5. We see that the first N + 1 coefficients of the N-colored Jones polynomial are the same as the first N + 1 coefficients of Φ 0. In [GL11], Garoufalidis and Le independently proved that the head and tail of the colored Jones polynomial exist for alternating knots while proving (for alternating knots) a stronger version of this stability. In particular, they showed the following property displayed below for the figure eight knot holds for all alternating links: 4
Φ 0 1-1 -1 0 0 1 0 1 0 0 0 0-1 0 0-1 N = 3 1-1 -1 0 2 0-2 0 3 0-3 0 3 0-3 0 N = 4 1-1 -1 0 0 3-1 -1-1 -1 5-1 -2-2 -1 6 N = 5 1-1 -1 0 0 1 2 0-2 -1-1 1 3 1-2 -3 Since we know all of Φ 0, we can subtract it from the shifted colored Jones polynomials. This gives us N + 1 leading zeros. 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. Garoufalidis and Le proved that we can continue this process indefinitely and the sequences, Φ n, will continue to stabilize. Φ 1 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 I call the sequence Φ 1 the neck of the tail or the tailneck of the colored Jones polynomial of the figure 8 knot. m 1 m 2 m 3 Figure 3: A trefoil knot with its checkerboard graph. I calculated the tailneck of all three strand pretzel knots with negative twists in each region see Figure 3. The m i represent the number of crossings in each section. As it is drawn, each m i = 1. (If m 1 = 2 and the others are 1, we get the figure 8 knot.) Theorem 3.1. 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 ) +, i.e. the pentagonal numbers plus the partial sum of 1 q the pentagonal numbers, if two m i = 1 and one is 2 or more. n=1 (1 qn n=1 ) + 2 (1 qn ), i.e. the pentagonal numbers plus the 2 times the 1 q partial sum of the pentagonal numbers, if one m i = 1 and two are 2 or more. 5
n=1 (1 qn n=1 ) + 3 (1 qn ), i.e. the pentagonal numbers plus the 3 times the 1 q partial sum of the pentagonal numbers, if all m i 2. This theorem gives us stabilization of length one less that that guarenteed in [GL11] but is consistent with the stabilization that appears to hold for these knots. 4 Future Work 4.1 The Middle Coefficients In my future work, I hope to continue to study these sorts of patterns in the coefficients of the colored Jones polynomial with an ultimate goal of gaining insight about the patterns I originally discovered. I would still like to understand the middle coefficients of the colored Jones polynomial better. In particular, I would like to find a better way to describe the pattern visible in the coefficients and prove the properties of the observed pattern. Since I have only been able to calculate the coefficients of the colored Jones for relatively simple knots and a relatively low number of colors, there may be other patterns for other knots. I would to understand why there is a period N oscillation in the coefficients. I hope to be able to relate the mth coefficient of the N colored Jones polynomial to the (m + N)th coefficient. In addition to looking at the oscillation, I would like to look at the magnitude of the highest coefficients in the oscillation and the maximum coefficient overall. Preliminary tests suggest that the growth rate of the maximum coefficient is related to the hyperbolic volume of the knot. I would like to find a way to study the maximum coefficient in order to see if this is in fact true. It seems that for the figure 8 knot, the maximum coefficient occurs in the middle (the constant term). I would like to prove this property and see what other knots it holds for. 4.2 Ways to Evaluate the Colored Jones Polynomial In order to prove the above conjectures, I would like to find new formulas for evaluating the colored Jones polynomial and gain a better understanding of other formulas and ways to calculate the colored Jones polynomial. I want to try to extend the formula I found for certain pretzel knots to other families of knots and see if there is an easier way to prove the formula in the case where r is even. I would also like to consider using matrices for computing the colored Jones polynomial. The hope is that this type of calculation will allow us understand how local changes (like adding a single crossing to a twist region) change to colored Jones polynomial, or the stabilized sequences related to it. 6
4.3 A Large Number of Twists While considering the stability of the colored Jones sequence, I found that having a large number of twists in each twist region leads to more stability. I would like to see if I can connect this idea of having a large number or twists to the work of Rozansky in [Roz10] which shows that we can think of the Jones-Wenzl idempotent as an infinite number of twists. I hope to find some connection between these two ideas. References [AD11] 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. [Arm11] C. Armond. The head and tail conjecture for alternating knots. ArXiv e-prints, December 2011. [Das] Dasbach, Oliver T.,Lin, Xiao-Song. A volumish theorem for the Jones polynomial of alternating knots. Pacific Journal of Mathematics, 231. [DL06] [GK12] [GL11] O. Dasbach and X.-S. Lin. On the head and the tail of the colored Jones polynomial. Compos. Math., 5:1332 1342, 2006. S. Garoufalidis and C. Koutschan. Irreducibility of q-difference operators and the knot 7 4. ArXiv e-prints, November 2012. S. Garoufalidis and T. T. Q. Le. Nahm sums, stability and the colored Jones polynomial. ArXiv e-prints, December 2011. [Mas03] G. Masbaum. Skein-theoretical derivation of some formulas of Habiro. Algebr. Geom. Topol., 3:537 556, 2003. [Mur10] H. Murakami. An Introduction to the Volume Conjecture. ArXiv e-prints, January 2010. [Roz10] L. Rozansky. An infinite torus braid yields a categorified Jones-Wenzl projector. ArXiv e-prints, May 2010. 7