On the tail of Jones polynomials of closed braids with a full twist

For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored Jones polynomial.


Introduction
The tail (resp. head) of a polynomial will denote the sequence of its lowest (resp. highest) degree terms, up to some specified length. In this note, we precisely determine the tail of the Jones polynomial for a closed n-braid with a full positive twist, and with up to n negative crossings. We also precisely determine the tails of the colored Jones polynomials for a closed positive n-braid with a full positive twist.
It is natural to consider quantum and geometric invariants of links that are closed braids with a full twist. For example, Lorenz links, all of which are closed positive braids with a full twist, dominate the census of the simplest hyperbolic knots, and their Jones polynomials are relatively simple [1,4]. The full twist arises as ±1 Dehn surgery on the braid axis, considered as an augmented unknot in S 3 . Hence, adding full twists is a natural geometric operation on links. On the other hand, the full twist is in the center of the braid group, so its image in any irreducible representation is a scalar. All known closed formulas for Jones polynomials of infinite link families essentially rely on this fact.
For any closed braid, we showed in [3] that after sufficiently many full twists on a subset of strands, the coefficient vector for any colored Jones polynomial decomposes into fixed blocks, separated by blocks of zeros that increase by a constant length for every twist. So once the non-zero blocks separate, they simply move apart unchanged with every additional full twist. In Theorem 1 below, we completely determine the first block for full twists on all strands. In this case, the first block separates after only one full twist. Dasbach and Lin [5] showed that for alternating knots, and more generally A-adequate knots, the first three coefficients in the tail of the normalized N -th colored Jones polynomials are independent of the color for N ≥ 3. In Corollary 4 below, for N > n − 1, we determine the tail of length n(N − 1) + 1 for the normalized N -th colored Jones polynomial of closed positive braids with a full twist. In fact, Theorem 3 implies that given M ≥ 2, for all colors N ≥ M , the coefficients of the tail of length M stabilize up to sign.
Dasbach and Lin also showed that the second coefficients of the head and tail together provide a linear bound for the hyperbolic volume of alternating knots. In contrast, the coefficients of the tail of length N as in Theorem 3 for closed positive braids with a full twist are all {−1, 0, 1}. Moreover, the tail of length n(N − 1) as in Corollary 4, also has coefficients only {−1, 0, 1}. These coefficients and the dependence on the braid index indicates that, for this class of knots, these tails by themselves are unrelated to the hyperbolic volume of the closed braid. For example, 3-braids can have unbounded hyperbolic volume, and the positive twisted torus knots T (p, q, 2, 2k) have bounded hyperbolic volume but unbounded braid index [2].
To state our main results, we adopt the following standard convention. Let V L (t) denote the Jones polynomial, such that Theorem 1 Let β be a n-braid of length c with negative crossings with 0 ≤ ≤ n and β = ∆ 2 n β , where ∆ 2 n is the positive full twist in the braid group B n . Then where p(β; t) and q(β; t) are polynomials in t.
The latter expression gives the tail of V β (t) of length n− +1. An interesting consequence is that the Jones polynomial satisfies a braid index constraint, which is a gap of length at least n − between the first two nonzero coefficients of (1 − t 2 ) · V β (t). If = 0 in Theorem 1, then β is a positive n-braid with a full twist. In this case, the MFW inequality [13,6] is sharp so the braid index of β is n. However, the gap between the first two nonzero coefficients of (1 − t 2 )V β (t) can be more than n. For example, if β = σ 2 2 σ 1 ∈ B 4 then β = ∆ 2 4 β and Another consequence of Theorem 1 is related to a conjecture of V. Jones [11], which remains open in general: The writhe w(β), which is the algebraic crossing number of β, is a topological invariant of β whenever n is the minimal braid index of β (see [14]). When the MFW inequality is sharp, the Jones conjecture is known to hold, which is the case for positive braids, = 0. If = 1, β is conjugate to a positive braid. For > 1, although we do not know when the MFW inequality is sharp, we can prove the Jones conjecture: Corollary 2 For β as in Theorem 1, Thus, if n is the minimal braid index of β, then w(β) is a topological invariant of β.
Let J N (L; t) be the colored Jones polynomial of L, colored by the N -dimensional irreducible representation of sl 2 (C), with the normalization The colored Jones polynomials are weighted sums of Jones polynomials of cablings, and the following formula is given in [12]. Let L (r) be the 0-framed r-cable of L; i.e., if L is 0-framed, then L (r) is the link obtained by replacing L with r parallel copies. (See below for a formula modified for other framings.) The normalized colored Jones polynomial Theorem 3 Let β be a positive n-braid of length c and β = ∆ 2 n β , where ∆ 2 n is the positive full twist in the braid group B n . Then where p N (β; t) and q N (β; t) are polynomials in t.

Corollary 4
If β is a positive n-braid of length c and β = ∆ 2 n β , then for N > n − 2, For N > n − 1, Corollary 4 determines the tail of length n(N − 1) + 1 for the normalized N -th colored Jones polynomial.
The Jones Slope Conjecture [9] relates boundary slopes of a knot K to the growth rate of the lowest and highest degrees of J N (K; t).
Corollary 5 Let j * (N ) denote the lowest degree of J N (β; t) for β as in Theorem 3. Then This proves part of the Jones Slope Conjecture for knots β.
Since β is A-adequate, Corollary 5 also follows from Example 9 in [7].

Remark 1
The AJ Conjecture [8] relates the recurrence properties of J N (K; t) to the A-polynomial of a knot K. The terms of J N (K; t) in Corollary 4 have exponents which are linear in the color N , so these terms give no information about the A-polynomial via the AJ-Conjecture.

Proof of Theorem 1
Generalizing the well-known formula for torus knots, the Jones polynomial of any torus link T (p, q) is given by the following sum, with d = gcd(p, q) [10].
For n = 2, the claim follows from (2), which in this case simplifies to Henceforth, let n > 2. The Jones polynomial V L (t) is obtained from the Kauffman bracket L by substituting t = A −4 and multiplying by (−A 3 ) −w to adjust for the writhe w of L.
We will show that the right-most part of the coefficient vector of the Kauffman bracket if n is even We will call this part of the Kauffman bracket the gap block. Multiplying by  (2), we obtain the Jones polynomial of (n, n) torus link = ∆ 2 n , which is the closure of the full twist in the braid group B n .
The Kauffman bracket of ∆ 2 n is obtained from (3) by substituing t = A −4 and to adjust for the writhe, multiplying by (−A) 3n(n−1) = A 3n 2 −3n . The first two terms of (1 − t 2 ) · V (t) change as follows: After dividing by 1 − A −8 we obtain a sum that depends on the parity of n. We see that the Kauffman bracket of the positive full twist on n-strands has a gap block with top degree n 2 + n − 2, and its smallest non-zero coefficient has degree n 2 − 3n ± 2, according to whether n is even or odd. However, when n is odd, the zero "term" is the first one in the gap block, so we say that the gap block has bottom degree n 2 − 3n − 2; i.e.
where q 2 (A) is a Laurent polynomial with degree strictly less than n 2 − 3n − 2 and a n = (1 + (−1) n )/2. We will show that adding the positive braid β to ∆ 2 n does not affect the gap block.
The Temperley-Lieb algebra T L n is closely related to the Jones polynomial. In the usual notation, T L n is the algebra over Z Let c = length(β ). For 0 ≤ ≤ n, the (unique) state which gives h 0 has B-smoothings, for which a − b = (c − ) − = c − 2 . We define q 1 (A) as follows: By Lemma 3, which is proved below, for i > 0 the highest power in ∆ 2 n h i is n 2 − 3n − 4. This implies the following: Lemma 1 If 0 ≤ ≤ n, the degree of q 1 (A) is at most c + n 2 − 3n − 6 + 2 .

Proof:
First, suppose = 0. We claim that to get k loops in any smoothing of β we need at least (k + 1) B-smoothings. Since β is a positive braid, every B-smoothing adds at most one loop, but the first B-smoothing does not result in any loops. Hence, (k + 1) B-smoothings (and the remaining A-smoothings) result in at most k loops.
If > 0, we claim that to get k loops in any smoothing of β we need at least (k + 1 − ) B-smoothings. As for a positive braid, (k + 1) smoothings that produce a cup-cap give at most k loops. But now, some of these smoothings could be A-smoothings at a negative crossing, so (k + 1 − ) B-smoothings (and the remaining A-smoothings) result in at most k loops.

It follows that
By Lemma 3, the highest power in ∆ 2 n h i(s) is n 2 − 3n − 4, so the degree of q 1 (A) is at most c + n 2 − 3n − 6 + 2 .
Let us compute the highest power of A after adjusting for the writhe. The writhe w = n(n − 1) + (c − ) − = n 2 − n + c − 2 . So after multiplying by (−A 3 ) −w , the highest power of A is To get the Jones polynomial, we substitute t = A −4 and multiply by (−A 3 ) −w . Hence we obtain the lowest power of t to be (−1) n+c+1 t (n−1) 2 +c−2 2 . This completes the proof of the first statement of Theorem 1.
To obtain the tail (without denominators), we note that 1 + tp(t) 1 − t 2 = 1 + tq(t), where q(t) is a polynomial. For simplicity, let = 0, but the proof is same in the other case. Given the polynomial p(t) from the proof above, we obtain the polynomial q(t) as follows: This completes the proof of Theorem 1. We will prove the claim by induction on the length of h as a product of e i 's. For the base case, h = e i . If E i denotes e i with its n − 2 through strands given a full right twist, then from Figures 1 and 2 we see that ∆ 2 n e i = A −6 E i .
Assuming the claim holds for h = e i1 . . . e ir−1 , we must show that it holds for h = h e ir . For the standard T L n basis, we have that i r is distinct from i 1 , . . . , i r−1 , so that cap(e ir ) cap(h ). This gives us two cases (see Figure 3):   Lemma 3 The highest power of A in ∆ 2 n h i for any i > 0 is less than or equal to n 2 − 3n − 4.

Proof:
Following the notation in Lemma 2, k = #cups in h i , so that 1 ≤ k ≤ [n/2]. By Lemma 2, ∆ 2 where H i also has k cups. The closure of H i will result in k cups paired with caps to produce loops, and k cups pulled through the full twist with a factor of A −6k , where 0 ≤ k, k ≤ k. So H i will have a full twist on m strands, where 0 ≤ m ≤ m. Thus, ∆ 2 The highest power of A is (−6k − 6k + 2k + (m ) 2 + m − 2), which is maximized when k = 0, k = k and m = m. Since 2k + m = n, we have m 2 + m − 2 − 4k = (n−2k) 2 +(n−2k)−2−4k. The function f (k) = (n−2k) 2 +(n−2k)−2−4k has an absolute minimum at k = n 2 + 3 4 and is decreasing for

Proof of Theorem 3
For any link diagram D, let D (r) denote its blackboard framed r-cable. Let D n denote the standard diagram of the closure of a full positive twist on n strands with a positive kink on each strand. Thus, D n is planar isotopic to ( ) (n) . Hence D (r) n = (( ) (n) ) (r) = ( ) (nr) = D nr .
For N ≥ 0, let S N (x) be the Chebyshev polynomials defined by For a link diagram D, let S N (D) be a linear combination of blackboard cablings of D.
We can define the colored Jones polynomial, as in equation (1), by a related expression in terms of the blackboard framing for D (see, e.g., [5]): As before, let β be a positive n-braid with c crossings, β = ∆ 2 n β , and L = β. Let D be a diagram of L given by the closure of β with a positive kink on every strand after the full twist. Note that the writhe of D, w(D) = n 2 + c. By Lemma 4, D (r) is the closure of the braid ∆ 2 nr β (r) with a positive kink on every strand following the full twist. Let . By equation (6), By equation (5) with = 0, where q 3 (A) and q 4 (A) are Laurent polynomials such that max deg(q 3 (A)) ≤ n 2 −3n−6+c and max deg(q 4 (A)) ≤ 0.
Note that d 1 (j) and d 2 (j) are both quadratic functions of j. As j increases from 0 to [N/2], they decrease and d 2 (j) > d 1 (j).
Proof: Note that n ≥ 3 and N ≥ 3. To obtain the tail (without denominators), we suppress β in the notation, This completes the proof of Theorem 3.