纽结和链环族的Jones多项式的零点

The colored Jones polynomials as vortex partition functions

We construct 3D $\mathcal{N}=2$ abelian gauge theories on $\mathbb{S}^2 \times \mathbb{S}^1$ labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in $\mathbb{S}^3$. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in $SU(2)$ Chern-Simons gauge theories on $\mathbb{S}^3$, and then our construction provides an explicit correspondence between 3D $\mathcal{N}=2$ abelian gauge theories and 3D $SU(2)$ Chern-Simons gauge theories. We verify, i…

关于各种量子不变量的渐近展开 II：扭转纽结的着色琼斯多项式在单位根 $e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{M}}}$ 和 $e^{\frac{2\pi\sqrt{-1}}{N}}$ 处

On the asymptotic expansions of various quantum invariants II: the colored Jones polynomial of twist knots at the root of unity $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{M}}}$ and $e^{\frac{2π\sqrt{-1}}{N}}$

本文研究扭转纽结的着色琼斯多项式在特定单位根处的渐近展开，给出公式并通过极限得到另一单位根处的展开。

Evolution for Khovanov polynomials for figure-eight-like family of knots

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t -> -1) limit.

Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial

In the paper, we describe the Drinfel'd double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with some application to knot invariants. The knot invariants we get is a generalization of the Jones polynomial, in particular, it recovers the Jones polynomial in rank $1$ case, while in rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.

Polynomial Invariants of Torus Knots and (p,q)-Calculus

We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY skein relation - the HOMFLY deformed (p,q)-numbers with one fermionic parameter.

伪幺正表示中纽结矩阵元导出的琼斯多项式

Jones polynomials from matrix elements of tangles in a pseudounitary representation

本研究回顾利用辫群与Temperley-Lieb代数矩阵表示计算琼斯多项式的方法，提出用伪幺正表示构造态、以矩阵元计算纽结不变量的方案，比迹方法更具应用价值，还证明了椒盐卷饼纽结的通用公式。

HOMFLY Polynomial Invariants of Torus Knots and Bosonic (q,p)-Calculus

For the one-parameter Alexander (Jones) skein relation we introduce the Alexander (Jones) "bosonic" q-numbers, and for the two-parameter HOMFLY skein relation we propose the HOMFLY "bosonic" (q,p)-numbers ("bosonic" numbers connected with deformed bosonic oscillators). With the help of these deformed "bosonic" numbers, the corresponding skein relations can be reproduced. Analyzing the introduced "bosonic" numbers, we point out two ways of obtaining the two-parameter HOMFLY skein relation ("bosonic" (q,p)-numbers) from the one-parameter Alexander and Jones skein relations (from the correspondin…

闭合4-辫子与Jones纽结猜想

Closed 4-braids and the Jones unknot conjecture

针对Jones纽结猜想中4股辫子情形的未解问题，研究发现一族以拧数为参数、可能对应非平凡纽结的双变量HOMFLY-PT多项式（其Jones多项式平凡），探究其性质并提出检验方法与多股推广思路。

Chern-Simons理论、纽结不变量、顶点模型和三流形不变量

Chern-Simons Theory, Knot Invariants, Vertex Models and Three-manifold Invariants

陈-西蒙斯理论是拓扑量子场论，为研究三维空间中的节点和链接提供了一个场论框架。这些都是可以精确而明确地求解的量子场论的罕见例子。Wilson连接算子的期望值产生一类连接不变量，其中最简单的是著名的Jones多项式。其他不变量比琼斯的不变量更强大。这些新的不变量是敏感的手征的所有结至少高达10交叉数不像琼斯是盲目的手征的其中一些。然而，所有这些不变量仍然不足以区分一类称为突变体的结。这些链接不变量可以交替地从二维顶点模型中获得。在谱参数的特定极限下，这种模型的$R$ -矩阵提供了辫群的表示。这反过来又用于构造链接不变量。利用Lickorish和Wallace的定理以及Kirby、Fenn和Rourke的定理，这些定理将三流形与框架链接上的外科手术联系起来，$S^3$中的这些链接不变量也可以用于构造三流形不变量。

Alexander and Jones polynomials of surgerized tst links

This paper is a continuation on the 2012 paper on "Cutting Twisted Solid Tori (TSTs)", in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to "surgerized tst links": recall that when performing $Φ^μ(n(τ), d(τ), M)$ on a tst $\langle τ\rangle$ where $M$ is odd, we obtain the tst link, $[Φ^μ(n(τ), d(τ), M)]$ that contains a trivial knot as one of its components. We then perform another operation $Φ^{μ'}(n(τ'), d(τ'), M')$ on that trivial knot to create a new link, which we call a "surgerized tst link" (stst link). If $M'$ is odd, we can repeat the …