Hitoshi Murakami
I follow Y. Yokota to explain how to obtain a tetrahedron decomposition of the complement of a hyperbolic knot and compare it with the asymptotic behavior of Kashaev's link invariant using the figure-eight knot as an example.
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We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten-Reshetikhin-Turaev invariants.
Kashaev's invariants for a knot in a three sphere are generalized to invariants of a knot in a three manifold. A relation between the newly constructed invariants and the hyperbolic volume of the knot complement is observed for some knots in lens spaces.
We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern--Simons invariant and the twisted Reidemeister torsion.
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he restricted his method to knot diagrams with certain conditions. Although these restrictions are general enough for any hyperbolic knots, we have to select a good diagram of the knot to apply his theory. In this article, we suggest more combinatorial way to calculate the complex vol…
In this article, we give a rough, and so not complete yet, proof of Kashaev's conjecture, that is, the volume conjecture for hyperbolic knots, where the hyperbolicity equations associated to knot diagrams appear as the stationary phase equations for Kashaev's invariants.
Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $\mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the fundamental group of its complement into $\mathrm{SL}_2(\mathbb{C})$; equivalently, we can think of $\mathrm{KR}(K)$ as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for $K$ a hyperbolic knot $\mathrm{KaRe}(K)$ can be viewed as a…
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the invariant trace field of the link. On the other hand, Yokota suggested a potential function of a hyperbolic knot, which gives the hyperbolicity equations and the complex volume of the knot. Applying the fact above to his theory, we explain that the potential function also gives all …
R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots $6_3$, $8_9$ and $8_{20}$ and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern-Simons invariants and propose a complexification of Kashaev's conjecture.
FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots
本文通过计算机实现证明了安德森-卡沙夫体积猜想对42,000个结的补集成立，并揭示了FAMED性质的新见解。
We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can…
The optimistic limit is the mathematical formulation of the classical limit which is a physical method to expect the actual limit by using saddle point method of certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. The modified version was easier to handle and more combinatorial than the original one. On the other hand, it was known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at the certain root of unity. The optimistic limit of…
I show various calculations of the limit of the colored Jones function for the figure-eight knot and confirm R. Kashaev's conjecture in this case.
In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a non-commutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, \cite{Kashaev-definition_delta_groupoid, def_anneau, Kashaev-Delta-groupoids_and_ideal_triangulations}. In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyse some examples on other…
We investigate the Reshetikhin--Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian S…
We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the figure 8 knot is false if one considers all the colors, but holds true if one restricts the colors to a subset of the set of positive integers.
On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot
本文利用 Ohtsuki-Yokota 鞍点方法，给出了沿扭转结做整系数 q-手术所得闭双曲 3-流形在单位根处的 Reshetikhin-Turaev 不变量的渐近展开公式。
The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
本文提出FAMED三角化概念，证明了双曲扭结和前42,000个双曲结的安德森-卡沙夫体积猜想，并揭示了Teichmüller TQFT中配分函数的指数衰减现象。
Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. W…
Extending Knot Polynomials of Braided Hopf Algebras to Links
本文研究了如何将Kashaev等人基于辫状霍普夫代数的多变量结多项式扩展到链环，并识别其中一些与已知链环不变量一致。
We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.
We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored Jones polynomial at the N-th root of unity, coincides with the Eichler integral of the character.
The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in $1/N$. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power series of a knot. We give examples that illustrate our method and its results.
Knots and Their Related $q$-Series
本文研究由纽结补的 Andersen-Kashaev 态积分因子分解得到的周期全纯函数矩阵，证明其定义的 PSL₂(ℤ)-上闭链与 Kashaev 不变量及 3D-指标相关，并提出矩阵值全纯量子模形式概念，以三个双曲纽结为例说明。
More on Kashaev limits of the quantum $A$-polynomials
本文研究对称着色 Jones 多项式的量子 $A$-多项式在双尺度准经典（Kashaev）极限下的行为，通过更多例子展示其分裂为两部分，并指出双曲体积作为积分常数未被完全确定。
1-loop equals torsion for two-bridge knots
本文通过结合Ohtsuki--Takata的工作与显式计算，证明了带尖双曲3-流形幂级数的1-圈项等于伴随Reidemeister挠率的猜想对双曲双桥结成立。
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.
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 $4π^2$.
A proof of the Teichmüller TQFT volume conjecture for $7_3$ knot
本文在 Andersen–Kashaev 的 Teichmüller TQFT 框架下，利用特定四面体分解证明了 $S^3$ 中双曲纽结 $7_3$ 补空间的重构体积猜想。
Perturbative invariants of cusped hyperbolic 3-manifolds
本文证明与理想三角剖分尖点双曲三维流形相关的形式幂级数是拓扑不变量，并猜想其与 Teichmüller TQFT 的 Kashaev 不变量及 Andersen–Kashaev 不变量在微扰论所有阶上一致。
We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order <=n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m>0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order <=m but arbitrarily large volume. We …
Summability for State Integrals of hyperbolic knots
该研究证明了双曲纽结$4_1$、$5_2$相关微扰级数的复苏性与Borel可求和性，提出可将Borel-Laplace重求和表示为Andersen-Kashaev状态积分组合的算法，明确了复苏结构并支持Stokes常数计算。
The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for diffeomorphisms of surfaces, backed up by numerical evidence. The conjecture involves isomorphisms between certain representations of the Kauffman bracket skein algebra of the surface, and the bulk of the article is devoted to the development of explicit methods to compute these isomorphisms. These combinatorial and algebraic techniques are exploited in two subseq…
We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm…
Torus Knots and Minimal Models Revisited : Rational VOA characters from non-hyperbolic knots
本文通过结合3D-3D和bulk-boundary对应关系，揭示了非双曲结补集的组合数据与有理VOA特征之间的联系，提出了新的Nahm-sum-like表达式和系统算法。
We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by nei…
The colored Jones polynomial does not determine the Links--Gould polynomial
本文证明彩色 Jones 多项式无法确定 Links--Gould 多项式，通过构造突变对和利用突变不变性差异，并推广至相关不变量族。
Knots, Perturbative Series and Quantum Modularity
本研究引入一种双曲纽结的矩阵值不变量，揭示其多重结构性质，提出精细化量子模性猜想并以三类简单双曲纽结验证，另有两篇续作拓展相关研究。
Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations
本文研究了Teichmüller TQFT中广义FAMED半几何三角化在半经典极限下的渐近行为，证明了分部函数的指数衰减及Jones函数的衰减率由Neumann-Zagier潜在函数控制，并验证了Andersen-Kashaev体积猜想。
We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,ρ)$, where $W$ is a compact closed oriented 3-manifold, $ρ$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms}, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the {\it idealization} of any fixed {\it $\Dd$-triangulation}; the $\Dd$-triangulations are sim…
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