Christine Ruey Shan Lee, Roland van der Veen
We exhibit an infinite family of knots with the property that the first coefficient of the n-colored Jones polynomial grows linearly with n. This shows that the concept of stability and tail seen in the colored Jones polynomials of alternating knots does not generalize naively.
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The Colored Jones Polynomial and the A-Polynomial of Knots
研究了纽结的色琼斯多项式与A-多项式之间的关系。我们建立了一个大类的2-桥结的AJ猜想（Garoufaldet），涉及有色琼斯多项式和A-多项式。一路上，我们还计算考夫曼括号绞链模块的所有2桥结。建立了交错纽结的色琼斯多项式的一些性质。
Random walks and the colored Jones function
可以证明，纽结的有色琼斯函数可以根据纽结的平面投影图上的路径计数来计算。在组合层次上，有色琼斯函数可以用它的权系统代替。我们给出了两个奇怪的公式的重量系统的有色琼斯函数：一个方面的永久矩阵相关联的和弦图，另一个方面的计数路径相交的和弦。
On the asymptotic expansion of the colored Jones polynomial for torus knots
在渐近展开的双曲规格的彩色琼斯多项式的环面结，我们确定不同的几何贡献，特别是陈-西蒙斯invaraint和Reidemeister扭转。
The colored Jones polynomial and the A-polynomial for twist knots
我们证明了对于扭结，A-多项式可以从色琼斯多项式的Masbaum公式中的被加数的递归得到。我们的结果支持了由S. Garoufalidis提出的AJ猜想。
A cabling formula for the colored Jones polynomial
我们证明了显式布线公式的有色琼斯多项式。作为应用，我们证明了体积猜想的所有零体积的纽结和链接，即所有的纽结和链接，从unknot获得的重复电缆和连接和。
We introduce colored Jones polynomials of nanowords and their categorification. We also prove the existence of a Khovanov-type bicomplex which has three grades.
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum representations.
We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a real number $ξ$. We show that if $ξ$ is sufficiently large, the colored Jones polynomial grows exponentially when $N$ goes to the infinity. Moreover the growth rate is related to the Chern-Simons invariant of the knot exterior associated with an $\mathrm{SL}(2;\mathbb{R})$ representation.
On the Head and the Tail of the Colored Jones Polynomial
有色琼斯多项式是三维空间中与纽结K相关联的一元劳伦多项式J（K，n）的级数。我们将证明，对于一个交替纽结K，J（K，n）的第一个和最后三个首项系数的绝对值是独立的n时，n是足够大的。样本节点的计算表明，这应该是正确的任何固定的首项系数的彩色琼斯多项式交替结。作为推论，我们得到了有色琼斯多项式的一个体积定理。
The asymptotic behavior of the colored Jones function of a knot and its volume
我展示了8字形结的有色琼斯函数极限的各种计算，并证实了R。Kashaev的猜想。
Mutation and the colored Jones polynomial
我们展示了具有相同多项式不变量和双曲卷的结的例子，具有不同的重合2电缆多项式和有色琼斯多项式，它们不是突变体。
The colored Jones polynomials and the simplicial volume of a knot
我们证明了色琼斯多项式集和广义亚历山大多项式集定义的Akutsu，Deguchi和Ohtsuki相交非平凡。此外，它表明，交集是（至少包括）的Kashaev的量子双对数不变量的链接。因此，卡沙耶夫的猜想可以重述如下：有色琼斯多项式决定双曲结的双曲体积。修改这一点，我们提出了一个更强的猜想：有色琼斯多项式确定任何结的单纯体积。如果我们的猜想是真的，那么我们可以证明一个纽结是平凡的当且仅当它的所有瓦西里耶夫不变量都是平凡的。
A bicomplex of Khovanov homology for colored Jones polynomial
我们构造了一个双复形来分类有色琼斯多项式。这项工作的动机是由安娜Beliakova和斯蒂芬Wehrli谁讨论了有色琼斯多项式的分类在他们的论文中提出的问题。
Asymptotics of the colored Jones function of a knot
对于三维空间中的一个纽结，可以将一列劳伦多项式与之相关联，其第$n$项是第$n$有色琼斯多项式。研究了当$\a$为固定复数且$n$趋于无穷大时，$n$次有色Jones多项式在$e^{\a/n}$上的值的渐近性质.当$\a$是一个足够小的复数时，我们分析了$1/n$中所有阶的渐近行为。此外，我们给出了$n$阶有色Jones多项式的系数和次数的上界，并应用于广义体积猜想的上界. Agol-Dunfield-Storm-W.Thurston的工作意味着我们的界是渐近最优的。此外，我们给出了当$\a$在$2 πi$附近时的广义体积猜想的结果。我们的证明使用至关重要的分圆扩展的有色琼斯函数，由于哈比罗。
In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation between logarithmic invariant and the hyperbolic volume of a cone manifold is also investigated.
On the colored Jones polynomials of ribbon links, boundary links and Brunnian links
哈比罗给出了Z[q，q^{-1}]的主理想，其中代数分裂链的有色琼斯多项式的某些线性组合取值。证明了带状环、边界环和Brunnian环的相同线性组合都包含在Z[q，q^{-1}]的由多个元素生成的小理想中.在本文中，我们证明了这些理想也是主的，每一个生成的分圆多项式的产品。
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of links. We also introduce three functors between categories of nanophrases, generalizations of links, and obtain their applications using colored Jones polynomials and their categorifications.
In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r \in \Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones…
On the Colored Jones Polynomial and the Kashaev invariant
我们表示的有色琼斯多项式的逆矩阵的量子行列式的条目中的$q$ -Weyl代数的$q$ -运营商，评估在平凡的功能（加上简单的替代）。证明了Kashaev不变量等价于行列式的另一种特殊赋值。我们还讨论了Kashaev不变量的行列式公式与纽结补的双曲体积的行列式公式之间的相似性，希望这将导致体积猜想的证明.
Categorification of the colored Jones polynomial and Rasmussen invariant of links
我们定义了一个家庭的正式Khovanov括号的彩色链接取决于两个参数。这些括号的同构类是有框有色链接的不变量。Bar-Natan函子应用于这些括号产生Khovanov和李同调论分类有色琼斯多项式。此外，我们研究的条件下，框架彩色链接配边诱导链转换我们的正式括号。我们猜想，对于特殊的参数选择，Khovanov和李同调论的有色链接是函子（最多的符号）。最后，我们将Rasmussen不变量扩展到链接并给出例子，其中该不变量是比多变量Levine-Tristram签名更强的sliceness障碍。
Difference and differential equations for the colored Jones function
纽结的有色琼斯函数是一列劳伦多项式。这是由TTQ展示的。Le和作者证明了这类序列是$q$ -完整的，即它们满足系数为$q$和$q^n$中Laurent多项式的线性$q$ -差分方程.我们从第一性原理证明了$q$ -完整序列在$q$ -Weyl环上产生模. Frohman-Gelca-LoFaro把后一个环与量子环面的偶函数环和环面的考夫曼括号绞链模等同起来。通过这种证明，我们研究了上述作者所提出的有色琼斯函数的正交理想、周边理想和递归理想之间的关系。本文的第二部分将有色琼斯函数的线性$q$ -差分方程转化为一族线性常微分方程，并进行了环展开。这种转换是WKB方法的一个版本，并且可能会在纽结的有色琼斯函数的渐近性问题上提供一些信息。最终版本。发表在《纽结理论及其分支》杂志上。
The colored Jones function is q-holonomic
一个多元多元函数称为完整的，如果粗略地说，它是由它的许多值通过许多多项式系数的线性递归关系确定的。蔡尔伯格是第一个注意到，抽象的概念完整性可以适用于验证，在一个系统的和计算机化的方式，组合身份之间的特殊函数。利用三维空间中连杆的色Jones函数的一般状态和定义，从第一性原理出发，证明了色Jones函数是一个q-真超几何函数的重和，因而它是q-完整的.我们通过计算机计算证明了我们的结果。
A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial
我们提出了一个猜想（已经被A. Kricker）关于纽结的Kontsevich积分的结构。我们描述它的价值的生成函数的外部边缘连接到封闭的三价图的数量。我们猜想，这些功能是有理函数的指数，他们的论点，他们的乘方的亚历山大-康威多项式的权力。这个猜想暗示了存在一个有色Jones（HOMFLY）多项式的q-1幂展开式，其系数是q^color的有理函数.我们展示了如何从有色SU（3）琼斯多项式的有理展开导出与θ-图相关的第一个Kontsevich积分多项式。
The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along a braid model of the colored Jones polynomial that was refined by Armond from the work of Huynh and Lê. The walk model gives rise to ordered words in a $q$-Weyl algebra which we address and study from multiple perspectives. We provide a highly optimized Mathematica implementation that exploits the modern features of the software. We include a performance ana…
In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.
The Colored Jones Function does not determine any colored Links--Gould polynomial
该研究证明有色琼斯多项式无法统一描述$\text{LG}^{(n)}$，通过$n=1$的情况及奇数绞合公式、范德蒙德论证，得出$n\boldsymbol{\text{≥}}2}$时两类纽结的卫星可被$\text{LG}^{(n)}$区分的结论。
Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.
On the relation between the A-polynomial and the Jones polynomial
在以前与Frohman和Lofaro的联合工作中，引入了纽结的A-多项式的非交换推广，由量子平面中多项式的非交换生成理想（非交换A-理想）组成。本文证明了纽结的非交换A-理想与它的三个色琼斯多项式一起决定了所有其它色琼斯多项式。此外，对于特定的纽结，如unknot和三叶形，非交换A-理想完全决定了所有的有色琼斯多项式。
SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
本文澄清和改进了色琼斯多项式的渐近性态与具有复规范群SL（2，C）的Chern-Simons规范理论之间的关系。精确的比较需要仔细理解一些微妙的问题，如有色琼斯多项式的归一化和Chern-Simons理论中极化的选择。解决这些问题，使我们能够超越体积猜想，并验证一些预测的行为的subleading条款的渐近展开的有色琼斯多项式。
A non-commutative formula for the colored Jones function
纽结的有色琼斯函数是对纽结及其平行线的琼斯多项式进行编码的洛朗多项式序列。它已经被理解为量子群的表象，威滕给出了一个内在的量子场论解释，将有色琼斯函数作为三维规范理论（陈-西蒙斯理论）的威尔逊环的期望值。我们提出的有色琼斯函数作为一个非交换的费米子分拆函数的逆的评价。这个结果是量子场论中常见的形式，即广义行列式的逆。我们的公式还揭示了亚历山大多项式和结的有色琼斯函数之间的直接关系，并立即暗示了被广泛研究的梅尔文-莫顿-罗赞斯基猜想，大约十年前由巴尔-纳坦和第一作者首次证明。
This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.
The "color" in the colored Jones polynomial is an integer parameter. In this paper, a periodic pattern of the values of the colored Jones polynomial at the second and the third roots of unity is found. If we substitute -1 to the colored Jones polynomial, the value is alternately 1 or the determinant of the given link. When it comes to substituting the primitive third root of unity, another periodic pattern appears.
Walks Along Braids and the Colored Jones Polynomial
利用有色琼斯多项式的Huynh和Le量子行列式描述，我们构造了有色琼斯多项式沿辫子行走的一个新的组合描述。然后，我们使用这种描述表明，对于一个结，这是一个积极的辫子的封闭，第N次有色琼斯多项式的前N个系数是平凡的。
We give an alternate expansion of the colored Jones polynomial of pretzel links which recovers the degree formula in arXiv:1807.00957. As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle pretzel knots.
We discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we find an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots. We also determine which knots have the same higher order stability.
On the Turaev-Viro endomorphism, and the colored Jones polynomial
通过应用由Blanchet、Habegger、Masbaum和Vogel构造的TQFT的一个变体，并使用Ohtsuki的构造，我们通过使用从K的外科手术表示获得的缠结来定义每个结K的模自同态。我们证明了它是强移位等价于与K相关联的Turaev-Viro自同态。在Viro之后，我们考虑在对纽结的经线和经线着色之后获得的自同态。我们表明，这些自同态的痕迹编码相同的信息作为色琼斯多项式的K在一个单位根。大部分的讨论是在更一般的设置无限循环覆盖的3-流形。
Optimistic limits of the colored Jones polynomials
我们发现，乐观的双曲结的有色琼斯多项式的极限与乐观的Kashaev不变量模$4π^2$的限制。
Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function
利用纽结图上的随机游动模型，研究了纽结的Alexander多项式和有色Jones多项式。在这种情况下，一个结的亚历山大多项式的逆起着一个有向加权图的Ihara-Selberg zeta函数的作用，用权重计算1-string链接上的随机游动的循环，该链接的闭包是所讨论的结。然后，有色琼斯多项式对单串链路的布线上的随机游动的“自回避”循环的加权族进行计数。作为亚历山大和有色琼斯多项式的这种解释的结果，当着色（或布线）参数趋于无穷大而权重参数趋于1时，重正化有色琼斯多项式的极限的计算立即导致了梅尔文-莫顿猜想的新证明，该猜想首先由Rozansky和Bar-Natan和Garoufalthy建立。
We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.
On the colored Jones polynomials of certain cable of the torus knots
本文研究了一类特殊纽结的色琼斯多项式在单位根上的渐近性态。我们证明了体积猜想所预言的极限为零。
We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial.
On the Potential Function of the Colored Jones Polynomial and the AJ conjecture
本文从参数化势函数的角度探讨了A-多项式与彩色琼斯多项式之间的关系，验证了AJ猜想与A-多项式潜力函数之间的联系。
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we verify Garoufalidis' Slope Conjecture and a conjecture of the authors for cables of a two-parameter family of closed 3-braids called 2-fusion knots.
Slopes and colored Jones polynomials of adequate knots
Garoufallen建立了纽结的边界斜率与其色琼斯多项式之间的关系。根据这一猜想，某些边界斜率可由有色琼斯多项式的次数序列来检测。我们验证这个猜想足够的结，一类，大大推广交替结。
We generalize the colored Jones polynomial to $4$-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.
Some limits of the colored Jones polynomials of the figure-eight knot
我们将研究8字形纽结的色琼斯多项式的渐近性态。特别是，我们将表明，对于某些限制，我们得到的体积的锥流形的奇点沿结。
The Head and Tail of the Colored Jones Polynomial for Adequate Knots
我们表明，头部和尾部功能的有色琼斯多项式的适当的链接是产品的头部和尾部功能的有色琼斯多项式的交替链接，可以读出小康的适当图的链接。我们应用这一点，以加强定理Kalfagianni，Futer和Purcell的纤维充分的联系。
Representations and the colored Jones polynomial of a torus knot
我们证明了对于环面纽结，SL（2;C）Chern-Simons不变量和SL（2;C）扭曲的Reidemeister挠率出现在有色Jones多项式的渐近展开中.这表明一个推广的体积猜想，涉及的渐近行为的有色琼斯多项式的一个结的体积补。
The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces
我们计算的极限的彩色琼斯多项式的8字结，并得出结论，在大多数情况下，他们确定的体积和陈-西蒙斯不变量的三个流形的德恩手术沿它。
Difference equation of the colored Jones polynomial for torus knot
证明了环面纽结T_{s,t}的N色Jones多项式满足二阶差分方程，对于T_{2,2m+1}情形，该差分方程可化为一阶差分方程.我们表明，A-多项式的环面结可以从这个差分方程。还构造了T_{2,2m+1}的有色Jones多项式的q-超几何型表达式。
On the volume conjecture of the colored Jones invariants with arbitrary colors
研究着色Jones不变量在链接补集超几何结构变形下的体积猜想，通过图八结和布洛美环的极限展示与超几何锥流形体积的关系。
There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.
We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.
A stability conjecture for the colored Jones polynomial
我们制定了一个稳定性猜想的系数的有色琼斯多项式的结，着色的不可约表示在一个固定的射线的一个简单的李代数，并验证它的所有环面结和所有的秩$2$的简单李代数。我们的猜想是由[Ga 2]中给出的$q$-完整多项式序列的次数和系数的结构定理以及\cite{GL2}中给出的交替纽结的有色琼斯多项式的稳定性定理激发的。我们说明我们的结果与样本计算。
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology
设S^3中的K是一个纽结，\widetilde{K}表示K在其双分支覆盖中的原像，即K的双分支覆盖，K（K）。对任意整数n > 1，证明了从K的镜像的约化n色Jones多项式到（K（K），\widetilde{K}）（当n为奇数）和（S^3，K # K）（当n为偶数）的结弗洛尔同调的Khovanov分类的谱序列的存在性.我们的结果的一个推论是，Khovanov的分类减少n-有色琼斯多项式检测unknot每当n>1。
We show that the family of colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with the action of the braid on configuration spaces. The sum is over the number of configuration points. Then we interpret this sum in terms of Poincare--Lefschetz duality intersection pairing between homology classes.
It is known that the colored Jones polynomial of a $+$-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the $+$-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for $+$-adequate links can be defined for $all$ links, and that it is trivial if and only if the link is non $+$-adequate.
The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot
我们表明，从8字形结的有色琼斯多项式的评估的渐近行为，我们可以提取陈-西蒙斯不变量和扭曲的Reidemeister扭转与表示的基本群的结补充二维复杂的特殊线性群。
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those…
Colored Jones invariants of links in S^3 # k S^2 X S^1 and the Volume Conjecture
我们使用基于Turaev阴影的状态和公式，将有色琼斯多项式的定义扩展到S^3 # k S^2 X S^1中的框架链接和三价图。然后，我们证明了体积猜想的自然扩张对于无限族双曲杆是成立的。
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