Daniel Amankwah
This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the Jones polynomial.
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A state generating is introduced to determine the Jones polynomial of a link. Formulae for two infinite families of knots are shown by applying this method, the second family of which are proved to be non-alternating. Moreover, the method is generalized to compute the Jones-Kauffman polynomial of a virtual link. As examples, formulae for one infinite family of virtual knots are given.
This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative results.
The Jones Polynomial of a Connect Sum is Multiplicative: A New Approach Via Trip Matrices
本研究采用行程矩阵方法，为连通和运算下琼斯多项式的乘法性提供了一种新的证明途径。
We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
A volume-ish theorem for the Jones polynomial of alternating knots
体积猜想声称一个纽结的双曲体积是由有色琼斯多项式决定的。本文的目的是用琼斯多项式而不是有色琼斯多项式来证明交替结点的体积定理：琼斯多项式的体积与某些系数之和的比值从上到下都由常数限定。此外，我们给出了实验数据的增长的双曲线体积和琼斯多项式的系数的关系，交替和非交替结。
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.
Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that for a knot which is the closure of a positive braid, the first N coefficients of the N-th colored Jones polynomial are trivial.
On the Jones polynomials of checkerboard colorable virtual knots
引入了虚拟链接和抽象链接的棋盘可染性概念。研究了虚链路和抽象链路的Jones多项式。证明了经典链环的琼斯多项式的某些性质对允许棋盘着色的虚链环也是有效的。
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 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 present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of non-isotopic links than the original Jones polynomial such as the Thistlethwaite link from the unlink with two components.
This article contains general formulas for Tutte and Jones polynomial for families of knots and links given in Conway notation.
On the Annihilating polynomial of the Colored Jones Polynomial for Some Links
研究着色Jones多项式在交换与非交换情况下的多项式与理想，提出链环版本的AJ猜想。
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.
We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.
We prove that there are only finitely many values of the Jones polynomial of quasi-alternating links of a given determinant. Consequently, we prove that there are only finitely many quasi-alternating links of a given Jones polynomial iff there are only finitely many quasi-alternating links of a given determinant.
We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4π^2$.
In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group $\mathbb{B}_6$ into a group of $5\times 5$ matrices. We also can calculate the Jones polynomial of the $2n$-plat presentations of knots by generalizing the method for the 6-plat presentations of knots.
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。
The Jones polynomial and the planar algebra of alternating links
这是一个众所周知的结果Thistlethwaite的琼斯多项式的非分裂交替链接是交替的。我们发现正确的推广这一结果的情况下，非分裂交替缠结。更具体地说：摘要缠结的Jones多项式在某个skein模中取值，本文描述了这个skein模中元素的一个交替条件，证明了这个交替条件由单交叉缠结的Jones不变量所满足，并证明了它通过适当的“交替”平面代数合成而保持不变。因此，所有交替缠结的琼斯多项式都满足这个条件。最后，在0-缠结的情况下，也就是链接，我们的条件等价于简单的交替的琼斯多项式的系数。
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.
Let $N=2n^2-1$ or $N=n^2+n-1$, for any $n\ge 2$. Let $M=\frac{N-1}{2}$. We construct families of prime knots with Jones polynomials $(-1)^M\sum_{k=-M}^{M} (-1)^kt^k$. Such polynomials have Mahler measure equal to $1$. If $N$ is prime, these are cyclotomic polynomials $Φ_{2N}(t)$, up to some shift in the powers of $t$. Otherwise, they are products of such polynomials, including $Φ_{2N}(t)$. In particular, all roots of unity $ζ_{2N}$ occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.
Purely cosmetic surgeries on knots with low-span Jones polynomials
本文证明了跨度至多11的非平凡Jones多项式纽结不允许纯美容手术，这是对纽结手术分类的重要进展。
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.
The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $\left(t-1\right)^{n}\left(t^{2}+t+1\right)\left(t^{2}+1\right)$, and (2) there exists a pair of two oriented knots which are $C_{n}$-equivalent such that the difference of the Jones polynomials for them equals $\left(t-1\right)^{n}\left(t^{2}+t+1\right)\left(t^{2}+1\right)$.
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones polynomial and the author's discovery of the bracket polynomial model for the Jones polynomial. We then describe some of the developments in knot theory that were inspired by the Jones polynomial and involve variations and generalizations of this invariant. The paper is written in the form of a personal odyssey and with the intent to show different mathemat…
Purely cosmetic surgeries on knots with low-span Jones polynomials, II
本文证明跨度至多14的非平凡Jones多项式纽结及编织指数至多4且奇数跨度Jones多项式的纽结均无纯美容手术，推广了先前结果并确定了跨度为15的多项式无限族。
In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $Δ_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial $Δ_K(t^d)$ where $d\in\mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in\{0,1,2,3\}$ these satellite knots have different Jones polynomials. For this purpose, I give rise to a formula for calculating the Jones polynomial of a satellite knot in terms of the Jones polynomials of its pattern and companion.
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。
Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.
On the Jones polynomial modulo primes
本文研究素数模下纽结 Jones 多项式的分布密度，证明大次数范围内其上界为 $4/p^7$，并据此分类跨度至多八的模二情形。
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 Jones and Alexander polynomials for singular links
我们扩展的状态模型的琼斯和亚历山大多项式的经典链接的状态模型的情况下，奇异链接的2变量多项式。此外，我们将这两个多项式与d+1变量的长奇异纽结正好d二重点。这些扩展可以检测长奇异节点的不可逆性。
We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links.
This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave differently from the original ones.
This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our 3-strand algorithm for the Jones polynomial is a special case of this generalization of the AJL algorithm. The present paper uses diagrammatic techniques to prove these results. The techniques of this paper have been used and will be used in the future in work with R. Marx, A. Fahmy, L. H. Kauffman, S. J. Lomonaco Jr.,A. Sporl, N. Pomplun, T. Schulte Herbruggen, J.…
We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For alternating Montesinos knots, these are the minimal and maximal boundary slopes.
We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that relates the asymptotic behavior of the colored Jones polynomial of a knot to the volume of the knot complement.
Nonalternating knots and Jones polynomials
虽然大多数结是非交替的，但纽结理论的现代研究似乎集中在交替结上。我们在这里考虑非交错纽结及其性质。具体来说，我们表明某些类的结有非平凡的琼斯多项式。
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。
The colored Jones polynomials and the Alexander polynomial of the figure-eight knot
体积猜想及其推广指出，一个纽结的有色琼斯多项式的某些估值的系列将指数增长，其增长率将与沿纽结通过Dehn手术获得的三流形的体积有关。在本文中，我们表明，对于8字结的级数收敛在某些情况下，极限等于其亚历山大多项式的逆。
It has been known that, the coefficients of the series expansion of the Jones polynomial evaluated at $e^x$ are rational valued Vassiliev invariants . In this article, we calculate minimal multiplying factor, λ, needed for these rational valued invariants to become integer valued Vassiliev invariants. By doing that we obtain a set of integer-valued Vassiliev invariants.
This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.
We show that if {L_n} is any infinite sequence of links with twist number tau(L_n) and with cyclotomic Jones polynomials of increasing span, then lim sup tau(L_n)=infty. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist--bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.
We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with …
Does the Jones polynomial detect the unknot?
我们解决的问题：是否存在一个平凡的结平凡的琼斯多项式？要找到这样一个纽结，几乎可以肯定的是，在布鲁表示的核心中找到一个非平凡的四股辫就足够了。我将描述一种计算机算法来搜索这样的辫子。
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.
We provide a new bound on the maximum degree of the Jones polynomial of a positive link with second Jones coefficient equal to $\pm 1$ or $\pm 2$. This builds upon the result of our previous work, in which we found such a bound for positive fibered links. We also show that each of these three bounds obstruct positivity for infinitely many almost-positive diagrams, allowing us to classify infinitely many knots as almost-positive.
We formulate a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra, and verify it for all torus knots and all simple Lie algebras of rank $2$. Our conjecture is motivated by a structure theorem for the degree and the coefficients of a $q$-holonomic sequence of polynomials given in [Ga2] and by a stability theorem of the colored Jones polynomial of an alternating knot given in \cite{GL2}. We illustrate our results with sample computations.
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.
Extreme coefficients of Jones polynomials and graph theory
我们发现家庭的总理纽结图与任意极端系数在他们的琼斯多项式。本文就这一问题提出了一些图论，推广了Yongju Bae和Morton的思想，并对他们论文中的一个问题给出了肯定的回答。
We show that from the asymptotic behavior of an evaluation of the colored Jones polynomial of the figure-eight knot we can extract the Chern--Simons invariant and the twisted Reidemeister torsion associated with a representation of the fundamental group of the knot complement to the two-dimensional complex special linear group.
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…
The Jones polynomial of collections of open curves in 3-space
本文提出了一种新的Jones多项式定义，用于描述三维空间中开放曲线集合的拓扑和几何复杂性。
Symmetric links and Conway sums: volume and Jones polynomial
我们得到的双曲型体积的周期链接和康威和交替缠结的界限。对于链接的康威和，我们也绑定的双曲卷的系数的琼斯多项式。
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