Research Statement Katherine Walsh - UCSD Mathematicsk3walsh/ResearchStatementExtended.pdf · Research Statement Katherine Walsh October 2013 Here, v 3 ˇ1:0149416 is the volume of

Research Statement Katherine Walsh October 2013

My research is in the area of topology, specifically in knot theory. The bulk of myresearch has been on the patterns in the coefficients of the colored Jones polynomial.The colored Jones polynomial is a knot invariant that assigns to each knot a sequenceof Laurent polynomials indexed by N ≥ 2 , the number of colors. For a knot K,denote the Nth term in this sequence JK,N(q), where N corresponds with the Ndimensional representation, i.e. we use the convention that when N = 2, we get theJones polynomial.

We usually think of the N -colored Jones polynomial as either the Jones polynomialof a linear combination of i-cablings of the knot for 0 ≤ i ≤ N−1 or as the evaluationin the Temperley-Lieb algebra of the knot diagram decorated with the N −1st Jones-Wenzl idempotent. In what follows, the colored Jones polynomial is normalized sothat its value on the unknot is 1.

One of the main open questions about the colored Jones polynomial is how torelate it to the geometry of the knot. One such relation is the following “hyperbolicvolume conjecture.”

Conjecture 1 ([Mur10], Kashaev-Murakami-Murakami). For any hyperbolic knot K,

2π limN→∞

log |JK,N(e2πi/N)|N

= vol(S3\K)

where JK,N(e2πi/N) is the normalized Colored Jones Polynomial of a knot K evaluatedat a N th root of unity and vol(S3\K) is the volume of the unique complete hyperbolicRiemannian metric on the knot complement.

The hyperbolic volume conjecture has been proved for torus knots, the figure-eightknot, Whitehead doubles of torus knots, positive iterated torus knots, Borromeanrings, (twisted) Whitehead links, Borromean double of the figure-eight knot, White-head chains, and fully augmented links (see [Mur10]). It is still open for other knotsand links.

In [Das], Dasbach and Lin related the first and last two coefficients of the originalJones polynomial to the the volume of the knot in the following way:

Theorem 0.1 (Dasbach, Lin). Volume-ish Theorem: For an alternating, prime, non-torus knot K let

JK,2(q) = anqn + · · ·+ amq

m

be the Jones polynomial of K. Then

2v8(max(|am−1|, |an+1|)− 1) ≤ Vol(S3 −K) ≤ 10v3(|an+1|+ |am−1| − 1).

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Here, v3 ≈ 1.0149416 is the volume of an ideal regular hyperbolic tetrahedron andv8 ≈ 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 Polyno-mial where also the first and last two coefficients of the N -colored Jones polynomialfor all N and noticed that the first and last N coefficents of the N -colored Jones poly-nomial seemed to be the same, up to sign, as the first N coefficients of the k-coloredJones polynomial for all k > N . We will discuss this further in section 3. These typesof theorem lead us to begin looking more deeply in to what the coefficients of thecolored Jones polynomial can tell us about the knot.

1 Patterns in the Coefficients of the Colored Jones

Polynomial

When studying the coefficients of the colored Jones polynomial, I first looked atpatterns in the entire set of coefficients. To be able to visualize these patterns, Iused a formula initially proved by Habiro and reproved by Masbaum in [Mas03] tocalculate the colored Jones polynomial of the figure 8 knot and twist knots and thenplotted the coefficients of these polynomials. The plot of the coefficients for the 95th

colored Jones polynomial of the figure 8 knot is below. (The plot has the degree ofthe term on the x−axis and the coefficient on the y−axis. Degrees were shifted bymultiplying by qM for some M so that all the degrees were positive.)

We see the same basic shape in other knots as well. Below is a similar plot forthe 30th colored Jones polynomial of knot 52.

This led me to the following conjectures about the basic shape of the plot of thecoefficients of the N th colored Jones polynomial.

1. In the middle, the coefficients of JK,N are approximately periodic with periodN .

2. There is a sine wave like oscillation with an increasing amplitude on the firstand last quarter of the coefficients.

3. We can see that the oscillation persists throughout the entire polynomial. Theamplitude starts small, grow steadily and then levels off in the middle and thengoes back down in a similar manner.

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5000 10 000 15 000

-3 ´ 1012

-2 ´ 1012

-1 ´ 1012

1 ´ 1012

2 ´ 1012

3 ´ 1012

Figure 1: Coefficients of the 95th Colored Jones Polynomial for the Figure Eight Knot

500 1000 1500 2000

-150 000

-100 000

-50 000

50 000

100 000

150 000

Figure 2: Coefficients of the 30th Colored Jones Polynomial for the Knot 52

I also looked at the growth rate of the maximum coefficients of each colored Jonespolynomial of a knot. The maximum coefficients of the polynomials seemed to growexponentially at a rate related to the hyperbolic volume of the knot.

Much of my research has been centered on trying to gain insight on where thesepatterns come from. This first led me to use the techniques from [Mas03] to find a

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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 coloredJones polynomial for large values of N . This formula is discussed in the next section.I also was led to look at the current work studying the first and last coefficients ofthe colored Jones polynomial, known as the head and tail of the polynomial and lookat the higher order stability of the coefficients. An overview of this research and howI have related it to my main questions is presented in the third section. In the lastsection, I explain how I hope to extend the results presented and future questions Ihope to answer.

2 A formula for the Colored Jones Polynomial of

a (1, r − 1, 2p− 1 pretzel knots

c1 c2 c3 cn

Figure 3: A (c1, c2, . . . , cn) Pretzel Knot. A box with a ci represents ci half twists.

A pretzel knot or link is usually described by P(c1, c2, . . . cn) where each ci is aninteger representing the number of half twists within that section of the knot. Thesetwisted parts are drawn vertically. Positive ci correspond with positive half twists,while negative ci correspond with negative half twists. See Figure 3.

We consider pretzel knots of the form P (1, 2p − 1, r − 1). Many of the knotswith a small number of crossings can be expressed as a pretzel knot of this form. Inparticular, we can express the knots in the Table 1 below as a (1, r−1, 2p−1) pretzelknot.

Following the techniques of Masbaum in [Mas03] we can find the following formulafor the colored Jones polynomial.

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Knot Twists Pretzel Notation (p,r)

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)91 (1,9,0) (5,1)92 4 (1,1,7) or (1,7,1) (1,8) or (4,2)95 (1,3,5) or (1,5,3) (2,6) or (3,4)

Table 1: A table of knots that can be expressed as a (1, r − 1, 2p− 1) pretzel knot.

Theorem 2.1. A pretzel knot of the form Kp,r = P (1, 2p− 1, r − 1) has the coloredJones polynomial

JN(Kp,r, a2) =

∑N−1n=0 c

′n,p

[N+nN−1−n

]µ∗n∑n

k=0 δ(2k;n, n)r 〈2k〉〈n,n,2k〉

([k]!)2

[2k]!{2n+1}!{n}!{1}

=∑N−1

n=0 (−1)n[N+nN−n−1

]c′n,p

{2n+1}!{n}!{1}

1(a−a−1)2n

∑nk=0 (−1)k(r+1) [2k+1]

[n+k+1]![n−k]!µr/22k .

Here

c′n,p =1

(a− a−1)nn∑k=0

(−1)kµp2k[2k + 1][n]!

[n+ k + 1]![n− k]!,

where µi = (−1)iAi2+2i and,

{n} = an − a−n, [n] =an − a−n

a− a−1[n

k

]:=

[n]!

[k]![n− k]!.

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Corollary 2.2. When r is even this reduces to

Jn(Kp,r, a2) =

N−1∑n=0

(−1)n[

N + n

N − n− 1

]c′n,p{2n+ 1}!{1}

c′n,r/2.

Corollary 2.3. When r is odd this reduces to

Jn(Kp,r, a2) =

N−1∑n=0

(−1)nµ4pn c′n,p

[N + n

N − 1− n

]{2n+ 1}!{n}!(a− a−1)2n{1}

n∑k=0

µ2k

r2

[2k + 1]

[n+ k + 1]![n− k]!

The formula for the case where r is even was independently proven by Garoufalidisand Koutschan in [GK12]. Using this formula, we are able to more quickly calculatethe 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 existsif the first N coefficients (of the highest order terms) of the N th polynomial in thesequence are the same as the first N coefficients of the kth polynomial for all k ≥ N .The tail of the sequence of polynomials, if it exists, is the stabilized sequences of thecoefficients of the lowest terms.

In [DL06, AD11, Arm11], Dasbach and Armond proved that the head and tail ofthe colored Jones exist for alternating and adequate knots and depend on the reducedcheckerboard graphs of the knot diagrams.

To obtain the reduced checkerboard graph given an alternating diagram of a knot,we assign a (gray/white) checkerboard coloring the faces in the diagram. We thenplace a vertex in each of the gray colored regions and draw an edge between verticesfor every crossing between the corresponding regions. Alternatively, we can start byplacing a vertex in every white region to get the dual graph. If, when moving along anedge, the overcrossing starts on the right of the edge and ends on the left, this graphis the A-checkerboard graph. If the overcrossing goes from the left to the right, thegraph is the B-checkerboard graph. See Figure 4. To get the reduced checkerboard,we can replace parallel edges in the graph, i.e. multiple edges between the samevertices, with a single edge.

In [GL11], Garoufalidis and Le independently proved that the head and tail of thecolored Jones polynomial exist for alternating knots while proving (for alternating

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(a) A diagram of61

(b) 61 with acheckerboardcoloring

(c) The A-checkerboardgraph

(d) The B-checkerboardgraph

Figure 4: The Knot 61 and its associated graphs.

knots) a stronger version of this stability. In particular, they defined the property ofk-stability for a sequence of polynomials as follows:

Definition 3.1. Suppose fn(q), f(q) ∈ Z((q)), i.e. fn(q) and f(q) are formal Laurentseries – series that can be written as

∑n≥N Anx

n where an ∈ Z. We write that

limn→∞

fn(q) = f(q)

if

• there exists C such that mindegq(fn(q)) ≥ C for all n, and

• for each j, there exist Nj such that for all n > Nj.

fn(q)− f(q) ∈ qjZ[[q]]

In particular, coeff(fn(q), qj) = coeff(f(q), qj).

Definition 3.2. A sequence (fn(q)) ∈ Z[[q]] is k-stable if there exist Φj(q) ∈ Z((q))for j = 0, . . . , k such that

limn→∞

q−k(n+1)

(fn(q)−

k∑j=0

Φj(q)qj(n+1)

)= 0.

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We call Φk(q) the k-limit of (fn(q)). We say that (fn(q)) is stable if it is k-stable forall k.

For example, a sequence (fn(q)) is 3-stable if

limn→∞

q−3(n+1)(fn(q)−

(Φ0(q) + q(n+1)Φ1(q) + q2(n+1)Φ2(q) + q3(n+1)Φ3(q)

))= 0.

The property of the head and tail existing for the colored Jones Polynomial of aknot is the same as the colored Jones sequence being 0−stable. In addition, in [GL11],Garoufalidis and Le proved the following theorem about higher order stability.

Theorem 3.3. For every alternating link K, the sequence (JK,n(q)) is stable and itsassociated k-limit ΦK,k(q) can be effectively computed from any reduced alternatingdiagram D of K.

I hope to be able to use this stabilization to extend the work done on the first andlast coefficients to the patterns in the middle that I originally observed. In particular,if I can find what the stabilized sequences are, I could extract which parts of thesesequences contributed to the maximum coefficient, or to other coefficients I wantedto study.

For example, for the figure 8 knot, we know the that first coefficients stabilize tothe pentagonal number sequence. By this, I mean that for the figure 8 knot,

Φ0 =∞∏n=1

(1− qn) =∞∑

k=−∞

(−1)kqk2(3k−1).

In the table below, I have listed out the first 16 coefficients of the N-colored Jonespolynomial 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 + 1coefficients of Φ0.

Φ0 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 · · ·

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

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Φ0 1 -1 -1 0 0 1 0 1 0 0 0 0 -1 0 0 -1 · · ·N = 3 0 0 0 0 2 -1 -2 -1 3 0 -3 0 4 0 -3 1 · · ·N = 4 0 0 0 0 0 2 -1 -2 -1 -1 5 -1 -3 -2 -1 7 · · ·N = 5 0 0 0 0 0 0 2 -1 -2 -1 -1 1 4 1 -2 -2 · · ·

Shifting these sequences back so that they start with a non-zero term, we can seethat 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 · · ·

I call the sequence Φ1 the “neck of the tail” or the “tailneck” of the colored Jonespolynomial of the figure 8 knot.

m1

m2m3

Figure 5: A trefoil knot with its checkerboard graph.

I calculated the tailneck of all three strand pretzel knots with negative twists ineach region. For knots in this family, the B-checkerboard graph is a three cycle.These knots can be drawn like the trefoil in Figure 5, except we will have morecrossings below the pictured crossings (and thus more parallel edges before we reducethe graph). The mi represent the number of crossings in each section. As it is drawn,each mi = 1. (If m1 = 2 and the others are 1, we get the figure 8 knot.)

Theorem 3.4. The tailneck of knots with reduce to the three cycle is:

•∏∞

n=1(1 − qn), i.e. the pentagonal numbers sequence, if all mi = 1 (The onlyknot satisfying this is the trefoil).

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•∏∞

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 mi = 1 and one is 2 or more.

•∏∞

n=1(1 − qn) + 2∏∞

n=1(1−qn)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.

•∏∞

n=1(1 − qn) + 3∏∞

n=1(1−qn)1−q , i.e. the pentagonal numbers plus the 3 times the

partial sum of the pentagonal numbers, if all mi ≥ 2.

This theorem gives us stabilization of length one less that that guarenteed inTheorem 3.3 but is consistent with the stabilization that appears to hold for theseknots.

4 Future Work

4.1 The Middle Coefficients

In my future work, I hope to continue to study these sorts of patterns in the coefficientsof the colored Jones polynomial with an ultimate goal of gaining insight about thepatterns I originally discovered. I would still like to understand the middle coefficientsof the colored Jones polynomial better. In particular, I would like to find a betterway to describe the pattern visible in the coefficients and prove the properties ofthe observed pattern. Since I have only been able to calculate the coefficients of thecolored Jones for relatively simple knots and a relatively low number of colors, theremay be other patterns for other knots.

I would to understand why there is a period N oscillation in the coefficients. Ihope to be able to relate the mth coefficient of the N colored Jones polynomial tothe (m+N)th coefficient.

In addition to looking at the oscillation, I would like to look at the magnitudeof the highest coefficients in the oscillation and the maximum coefficient overall.Preliminary tests suggest that the growth rate of the maximum coefficient is relatedto the hyperbolic volume of the knot. I would like to find a way to study the maximumcoefficient 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 toprove this property and see what other knots it holds for.

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4.2 Ways to Evaluate the Colored Jones Polynomial

In order to prove the above conjectures, I would like to find new formulas for evaluat-ing the colored Jones polynomial and gain a better understanding of other formulasand ways to calculate the colored Jones polynomial. I want to try to extend theformula I found for certain pretzel knots to other families of knots and see if there isan 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 poly-nomial. The hope is that this type of calculation will allow us understand how localchanges (like adding a single crossing to a twist region) change to colored Jones poly-nomial, or the stabilized sequences related to it.

4.3 A Large Number of Twists

While considering the stability of the colored Jones sequence, I found that havinga large number of twists in each twist region leads to more stability. I would liketo see if I can connect this idea of having a large number or twists to the work ofRozansky in [Roz10] which shows that we can think of the Jones-Wenzl idempotentas an infinite number of twists. I hope to find some connection between these twoideas.

References

[AD11] C. Armond and O. T. Dasbach. Rogers-Ramanujan type identities and thehead and tail of the colored Jones polynomial. ArXiv e-prints, June 2011.

[Arm11] C. Armond. The head and tail conjecture for alternating knots. ArXive-prints, December 2011.

[Das] Dasbach, Oliver T.,Lin, Xiao-Song . A volumish theorem for the Jonespolynomial of alternating knots. Pacific Journal of Mathematics, 231.

[DL06] O. Dasbach and X.-S. Lin. On the head and the tail of the colored Jonespolynomial. Compos. Math., 5:1332–1342, 2006.

[GK12] S. Garoufalidis and C. Koutschan. Irreducibility of q-difference operatorsand the knot 7 4. ArXiv e-prints, November 2012.

[GL11] S. Garoufalidis and T. T. Q. Le. Nahm sums, stability and the colored Jonespolynomial. ArXiv e-prints, December 2011.

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[Mas03] G. Masbaum. Skein-theoretical derivation of some formulas of Habiro. Al-gebr. 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 pro-jector. ArXiv e-prints, May 2010.

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Research Statement Katherine Walsh - UCSD Mathematicsk3walsh/ResearchStatementExtended.pdf · Research Statement Katherine Walsh October 2013 Here, v 3 ˇ1:0149416 is the volume of