The Head and Tail Conjecture for Alternating Knots
Cody Armond
Mathematics Department
Louisiana State University
Baton Rouge, Louisiana
Abstract.
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques from skein theory.
1991 Mathematics Subject Classification:
1. Introduction
The normalized colored Jones polynomial JN,L(q) for a link L is a sequence of Laurent polynomials in the variable q1/2, i.e. JN,L∈Z[q1/2,q−1/2]. This sequence is defined for N≤2 so that J2,L(q) is the ordinary Jones polynomial, and JN,U=1 where U is the unknot. For links L with an odd number of components (including all knots), JL,N is actually in Z[q,q−1]. For links with an even number of components, q1/2JL,N∈Z[q,q−1].
In [2] and [3] Oliver Dasbach and Xiao-Song Lin showed that, up to sign, the first two coefficients and the last two coefficients of JN,K(q) do not depend on N for alternating knots. They also showed that the third (and third to last) coefficient does not depend on N so long as N≥3. This and computational data led them to believe that the k-th coefficient does not depend on N so long as N≥k. The goal of this paper is to prove this conjecture for all alternating links. It is also known that this property of JN,L(q) does not hold for all knots. In [1], with Oliver Dasbach, we examined the case of the (4,3) torus knot for which this property fails.
Definition.
For two Laurent series P1(q) and P2(q) we define
P1(q)˙=nP2(q)
if after multiplying P1(q) by ±qs1 and P2(q) by ±qs2, s1 and s2 some powers, to get power series P′1(q) and P′2(q) each with positive constant term, P′1(q) and P′2(q) agree modqn.
For example −q−4+2q−3−3+11q˙=51−2q+3q4.
Another way of phrasing the above definition is that P1(q)˙=nP2(q) if and only if their first n coefficients agree up to sign.
In [1] we defined two power series, the head and tail of the colored Jones polynomial HL(q) and TL(q).
Definition.
The tail of the colored Jones polynomial of a link L – if it exists – is a series TL(q), with
TL(q)˙=NJL,N(q), for all N
Similarly, the head HL(q) of the colored Jones polynomial of L is the tail of JL,N(q−1), which is equal to the colored Jones polynomial of the mirror image of L.
Note that TL(q) exists if and only if JL,N(q)˙=NJL,N+1(q) for all N. For example, for the first few colors N the colored Jones polynomial of the knot 62 multiplied by q2N2−N−1 is
This is exactly the property conjectured by Dasbach and Lin to hold for all alternating knots, and the subject of the main theorem of this paper.
Theorem 1.
If L is an alternating link, then JL,N(q)˙=NJL,N+1(q).
Because the mirror image of an alternating link is alternating, this theorem says that the head and the tail exists for all alternating links. Theorem 1 was simultaneously and independently proved by Stavros Garoufalidis and Thang Le in [4] using alternate methods.
We are also able to prove a more general theorem about A-adequate links.
Theorem 2.
If L is a A-adequate link, then JL,N(q)˙=NJL,N+1(q).
Because all alternating links are A-adequate, Theorem 2 implies Theorem 1.
1.1. Plan of paper
In section 2, we discuss definitions and basic results regarding adequate links and skein theory. In section 3, we present the main Lemma which is a slight generalization of a Lemma in [1] relating the lowest terms of the colored Jones polynomial to a certain skein theoretic graph. Finally, in section 4, we present the proofs of Theorems 1 and 2 using the graph discussed in section 3.
1.2. Acknowledgements
I would like to thank Oliver Dasbach for all his help and advise. I would also like to thank Pat Gilmer for teaching me all I know about skein theory.
2. Background
2.1. Alternating and Adequate
Given a link diagram D there are two ways to smooth each crossing, described in Figure 1. A state of the diagram is a choice of smoothing for each crossing. Two states are particularly important when dealing with the colored Jones polynomial; they are the all-A state SA and the all-B state SB. The all-A (respectively all-B) state is the state for which the A (B) smoothing is choosen for every crossing.
For a state S we can build a graph GS called the state graph for S. The graph GS has vertices the circles in S, and edges the crossings in D. Each edge connects the two vertices corresponding to the two circles that the crossing meets.
Definition.
A link diagram is A-adequate (B-adequate) if the state graph for SA (SB) has no loops.
A link diagram is adequate if it is both A and B-adequate.
A link is adequate if it has an adequate diagram.
The most important property of A-adequate diagrams is that the number of circles in SA is a local maximum. In other words, any state that has only a single B smoothing will have one fewer circle than the all-A smoothing. Similarly for B-adequate diagrams, that the number of circles in SB is a local maximum.
It is a well-known fact that all alternating links are adequate links. In particular, a reduced alternating diagram, that is an alternating diagram without any nugatory crossings, is an adequate diagram.
Another important fact about adequate diagrams is that parallels of A-adequate diagrams are also A-adequate diagrams. Given a diagram D, the r-th parallel of D denoted Dr is the diagram formed by replacing D with r parallel copies of D.
2.2. Skein Theory
For a more detailed explanation of skein theory, see [5] or [6]
The Kauffman bracket skein module, S(M;R,A), of a 3-manifold M and ring R with invertible element A, is the free R-module generated by isotopy classes of framed links in M, modulo the submodule generated by the Kauffman relations:
If M has designated points on the boundary, then the framed links must include arcs which meet all of the designated points.
In this paper we will take R=Q(A), the field of rational functions in variable A with coefficients in Q. As we are concerned with the lowest terms of a polynomial, we will need to express rational functions as Laurent series. This can always be done so that the Laurent series has a minimum degree.
Definition.
Let f∈Q(A), define d(f) to be the minimum degree of f expressed as a Laurent series in A.
Note that d(f) can be calculated without referring to the Laurent series. Any rational function f expressed as PQ where P and Q are both polynomials. Then d(f)=d(P)−d(Q).
We will be concerned with two particular skein modules: S(S3;R,A), which is isomorphic to R under the isomorphism sending the empty link to 1, and S(D3;R,A), where D3 has 2n designated points on the boundary. With these designated points, S(D3;R,A) is also called the Temperley-Lieb algebra TLn.
We will give an alternate explanation for the Temperley-Lieb algebra. First, consider the disk D2 as a rectangle with n designated points on the top and n designated points on the bottom. Let TLMn be the set of all crossing-less matchings on these points, and define the product of two crossing-less matchings by placing one rectangle on top of the other and deleting any components which do not meet the boundary of the disk. With this product, TLMn is a monoid, which we shall call the Temperley-Lieb monoid. It has generators hi as in Figure 2, and following relations:
hihi=hi
hihi±1hi=hi
hihj=hjhi if |i−j|≥2
Any element in TLn has the form ∑M∈TLMncMM, where cM∈Q(A). Multiplication in TLn is slightly different from multiplication in TLMn, because hihi=(−A2−A−2)hi in TLn.
There is a special element in TLn of fundamental importance to the colored Jones polynomial, called the Jones Wentzl idempotent, denoted f(n). Diagramatically this element is represented by an empty box with n strands coming out of it on two opposite sides. By convention an n next to a strand in a diagram indicates that the strand is replaced by n parallel ones.
With
Δn:=(−1)nA2(n+1)−A−2(n+1)A2−A−2
and
Δn!:=ΔnΔn−1…Δ1 the Jones-Wenzl idempotent satisfies
If M∈TLMn, define fM∈R as the coefficient of M in the expansion of the Jones-Wenzl idempotent. Thus f(n)=∑M∈TLMnfMM. If e is the identity element of TLMn, then fe=1.
Lemma 3.
If M∈TLMn, then d(fM) is at least twice the minimum word length of M in terms of the hi’s.
Proof.
This follows easily from the recursive definition of the idempotent by an inductive argument. The only issue is that terms of the form