Higher Skein Modules
Jorgen Ellegaard Andersen, Vladimir Turaev
我们引入了更高的链的绞链模推广的康威绞链模。我们表明，这些模块是紧密相连的HOMFLY多项式。
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
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Fundamentals of Kauffman bracket skein modules
Skein模是基于结点（或位置）的代数拓扑的主要对象。本着与莱布尼茨相同的精神，我们将把我们的方法称为“代数位置”。“当我们观察绞链模的全景时，我们看到，经过同调和同伦的起伏山丘，遥远的山脉-考夫曼括号绞链模，以及更远的基于其他量子不变量的绞链模。我们在这里集中在考夫曼括号绞模的基本性质;性质的基础上进一步发展的理论。特别是，我们考虑相对考夫曼括号绞链模块，我们分析绞链模块的I-丛表面。
In this paper we introduce the notion of admissible skein modules associated to an ideal in a pivotal category. We explain how these modules are generalizations of the Kauffman skein algebra and how they relate to renormalized quantum invariants coming from non-semisimple categories.
Finiteness and holonomicity of skein modules
本文通过引入边界，证明三维流形编织模的有限性，并利用编织转移双模和q-模拟方法，证明内部编织模的全纯性，包括有限生成与拉格朗日支撑条件。
Carrega has shown that the Kauffman bracket skein module of the 3-torus over the field of rational functions in the variable A can be generated by 9 skein elements. We show this set of generators is linearly independent.
We compute the Kauffman skein module of the complement of torus knots in S^3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2,C)-characters tensored with the ring of Laurent polynomials.
We compute the action of the $\mathfrak{gl}_2$-skein algebra of the torus on the $\mathfrak{gl}_2$-skein module of the solid torus. As a result, we show that the $\mathfrak{gl}_2$-skein modules of lens spaces is spanned by $\left(\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)\left(2\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)$ elements.
A Note on the Bar-Natan Skein Module
基于适当嵌入的曲面及其关系，我们引入了一个新的三维流形的skein模，该模由D.Bar-Natan提出，并由M. Khovanov修改。我们计算了一些流形的模的结构，包括塞弗特流形。
On the Kauffman bracket skein module of the quaternionic manifold
利用重耦合理论研究了Z[A，A^{-1}]上四元数流形的Kauffman括号skein模，通过反演所有的分圆多项式，得到了该流形的局部化的Kauffman括号skein模.我们证明了绞链模是由五个元素构成的。利用这些绞元的量子不变量和流形的Z_2同调，我们确定它们是线性无关的。
Skein Modules and Lattice Gauge Field Theory
本文的修正版本见q-alg/9710003。
Frobenius algebras and skein modules of surfaces in 3-manifolds
对于每个Frobenius代数，定义了嵌入在给定的3-流形中的曲面的skein模，并在边界上限定了一个指定的曲线系统。绞关系是局部的，并产生相应的拓扑量子场论的某个自然扩张的核。特别是3-球的skein模同构于Frobenius代数的基环。我们证明了一个表示定理的绞链模块与发电机不可压缩曲面着色的Frobenius代数的生成集的元素，并与确定的管道几何关系的流形和关系的代数。
Deformation of string topology into homotopy skein modules
研究了Chas和Sullivan的弦拓扑与Hoste和Przytycki的同伦skein模之间的关系。这提供了新的见解同伦绞模的结构和它们的意义在量子拓扑的框架。我们的结果可以被认为是Turaev和Goldman关于定向曲面上的交和绞理论的经典结果的所有可定向三维流形的弱推广。
Let Sigma be a closed oriented surface of genus g. We show that the Kauffman bracket skein module of Sigma x S^1 over the field of rational functions in A has dimension at least 2^{2g+1}+2g-1.
The Kauffman skein module of the connected sum of 3-manifolds
设k是包含可逆元α，s和\frac{1}{s-s^{-1}}的整环.若M是定向3-流形，设K（M）表示M在k上的Kauffman skein模.在Beliakova和Blanchet关于Birman-Murakami-Wenzl代数的工作的基础上，我们给出了双曲体的Kauffman skein模的一个“类幂等”基. Gilmer和Zhong研究了两个3-流形的连通和的Homflypt skein模，本文研究了Kauffman skein模的情形，证明了K（M_1 # M_2）同构于某个局部环上的K（M_1）张量K（M_2），其中M_1 # M_2是两个流形M_1和M_2的连通和.
Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over $S^2$.
We study the stated skein modules of marked 3-manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3-manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein al…
Type $C$ skein modules and transparent elements
本文利用$Sp(2n)$ webs研究skein模，在$C$型背景下定义Chebyshev多项式的多变量类似物，并基于图示论证与编织公式构造单位根处的透明元素。
On the Kauffman skein modules
设k是α，s中有理函数域的子环，其中包含α^{1}，α^{-1}，s^{1}，s^{-1}，.设M是紧致定向3-流形，K（M）表示M在k上的Kauffman skein模.则K（M）是由M中框架链环的合痕类模Kauffman skein关系生成的自由k-模。在α，s中的有理函数域k={Q}（α，s）的情形下，给出了实心环面的Kauffman skein模的基和边界上有两点的实心环面的相对Kauffman skein模的基.然后我们证明K（S^{1} \times S ^2）是由空链接生成的，即，K（S^{1} \times S^2）=k。
Skein lasagna modules and handle decompositions
本文研究四维流形及其边界链环的 skein lasagna 模，提出利用环柄分解表示该不变量的一般方法，并通过实例计算揭示其可能具有局部无限维性。
We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub-algebras of intertwiners for partic…
The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $Σ$ we provide an explicit spanning family for the skein modules $S(Σ\times S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their question about the dimension of $S(Σ\times S^1)$ being $2^{2g+1} + 2g -1$.
Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs
代数Situs是数学的一个分支，其根源在于琼斯的建设，他的多项式不变的联系和德林费尔德的工作量子群。它包括理论的量子不变量的节点和3-流形，代数拓扑的基础上结，运算，平面代数，q-变形，量子群，并重叠与代数几何，非交换几何和统计力学。基于结的代数拓扑学可以被描述为通过考虑流形中的链接（子流形）及其代数结构来研究流形的性质。这门学科的主要对象是绞链模，它是流形中链路的环境合痕类上的自由模通过适当选择局部（绞链）关系的等价物。在这堂课中，我们集中讨论三维流形的一个相对简单的例子--q-同伦的skein模。这个绞链模块已经有了理论的许多成分：代数结构，相关的李代数，量子化，状态模型.
4-dimensional Skein modules, Handle attachments, and Tangles
本文提出了一种四维 skein 模块在处理附件中的通用公式，基于 skein 模块在 $S^3 \times I $ 中的定义，并扩展了现有研究成果。
The Yang-Mills Measure in the Kauffman Bracket Skein Module
对于每个闭的可定向曲面F，我们构造了Kauffman括号skein模K_t（Fx [0，1]）上的一个局部的非同态不变迹.当|t|既不为0也不为1，并且在某个单位根处时，定义迹。在t =-1时，迹是F的基本群的SU（2）特征标簇上对辛测度的积分。
The Homflypt skein module of a connected sum of 3-manifolds
设M是定向3-流形，S（M）表示M的Homflypt skein模.我们证明了S（M_1）连通和M_2同构于S（M_1）张量S（M_2）模挠。事实上，我们证明了在一定的局部化环上，S（M_1）连通和M_2）同构于S（M_1）张量S（M_2）。我们证明了相似的结果也适用于相对skein模。如果M包含一个分离2-球面，我们给出了在特定的局部化环上M的某些相对skein模为零的条件。
In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a "Chebyshev-Frobenius homomorphism" between skein modules of marked 3-manifolds…
We calculate the Kauffman bracket skein module (KBSM) of the complement of all two-bridge links. For a two-bridge link, we show that the KBSM of its complement is free over the ring $\BC[t^{\pm 1}]$ and when reducing $t=-1$, it is isomorphic to the ring of regular functions on the character variety of the link group.
The Kauffman Bracket Skein Module at an irreducible representation
本文研究闭定向三维流形在非四倍数单位根处的 Kauffman 括号 skein 模，证明其在不可约表示对应极大理想处的局部化是一维自由模，并由此给出同调球 skein 模维数的下界及有理函数系数的维数计算。
A categorification of the skein module of tangles
我们推广了我们以前关于曲面Kauffman括号skein模的分类的工作，将我们的同调推广到曲面Fx [0，1]上圆柱体中的缠结.我们在D^3中的0-缠结和1-缠结的同源性与Khovanov连接同源性和约化的Khovanov连接同源性一致（直到归一化）。我们证明了我们的同调的基本性质。特别是，短的确切序列的同调绞相关缠结和Kunneth公式的张量积缠结。
Skein modules of closed 3 manifolds define line bundles over character varieties
本文利用Frobenius态射将闭3-流形在奇数阶单位根处的绞纱模解释为SL2特征概形上凝聚层的全局截面空间，并证明当特征概形约化时该凝聚层为线丛。
Skein Modules and the Noncommutative Torus
证明了环面上圆柱的Kauffman括号skein模嵌入为非交换环面的一个子代数。利用这一点，我们推导出很好的公式的琼斯-Wenzl幂等元和分析结构的考夫曼括号绞链模块的unknot作为一个模块的考夫曼括号绞链模块的圆柱体上的环面。
Skein modules
我们在本章（第九章）中描述了基于节点（或者更一般地基于嵌入对象的位置）构建代数拓扑的思想。也就是说，我们的基本构建块被认为是环境合痕（不是同伦或同调）。例如，一个应该开始从结在3流形，表面在4流形等，但我们的理论是，到现在为止，发展只有在案件中的链接在3流形，只有一眼对4流形。该理论的主要对象是一个绞链模，我们致力于本章主要描述绞链模在三维流形。H.庞加莱，在他的论文“分析situs”（1895年），抽象地定义了同源群开始正式线性组合的单纯，选择周期和划分他们的关系来自边界的想法背后绞模块是使用链接，而不是周期（在案件的3流形）。更确切地说，我们考虑的自由模产生的链接模适当选择（本地）绞关系。
A survey on skein modules via braids
本文综述了利用辫子和纽结代数计算三维流形 skein 模的最新进展，重点介绍了实心环面、透镜空间及 $S^1\ imes S^2$ 的 HOMFLYPT 与 Kauffman 括号 skein 模的辫子方法。
Stated skein modules of 3-manifolds and TQFT
本文推广Kauffman括号纽结模至带标记3-流形得到带标记纽结模，证明量子参数为1的根时连通和相关自然映射非单射，还推导其作为幺半群对称函子的性质及与霍奇同调的关系。
We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.
Skein modules at the 4th roots of unity
对于所有的3-流形M，当A=+1，-1时，通过把它们与SL（2，C）-特征标簇联系起来，计算了Kauffman括号skein模S（M，A）.我们将这种描述扩展到当A是1的4次根且M是曲面x [0，1]或有理同调球面（或其子流形）的情况。
We show that relations in Homflypt type skein theory of an oriented $3$-manifold $M$ are induced from a $2$-groupoid defined from the fundamental $2$-groupoid of a space of singular links in $M$. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental $2$-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.
The Kauffman bracket skein module of a twist knot exterior
我们计算的Kauffman括号绞链模的补充的扭结，发现它是自由的和无限维的。该基础由两个组件链接的电缆组成，其中一个组件是结的子午线。子午线的布线可以是任意大的，而其他组件的布线则受限于扭曲的数量。
We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product formulae from the theory of tilting modules of the quantum group $U_q(sl_2)$. We show that the simplest skein identities yield a short new proof of the existence of the Chebyshev-Frobenius homomorphism of Bonahon-Wong.
Skein theory, line defects, and quantum symmetric pairs
本文从带边张量范畴及其平衡辫子模范畴出发，构造带线缺陷三维流形的辫理论，证明量子对称对缺陷辫模的有限性，并在特定情形下建立与Z2等变辫理论的等价，进而构造Type AIII情形的C∨C DAHA模。
On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$
本文在 Laurent 多项式环上计算了非素流形 $(S^1 \ imes S^2) \# (S^1 \ imes S^2)$ 的 Kauffman 括号 skein 模，通过分析 handle sliding 关系子模给出合适基，并证明其不能分解为自由子模与挠子模的直和。
Presentations of homotopy skein modules of oriented 3-manifolds
本文提出了一种新的求绞链模表示的方法。对于同伦skein模的情况下，它将显示如何在一个3流形的拓扑反映在结构的模块。解决了q-同伦skein模的自由度问题，并计算了一个与环绕数有关的自然skein模.
This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by R. Kirby and P. Melvin that underlie the Reshetikhin-Turaev model for $SU(2)$ Chern-Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.
Derived skein module
提出面向定向三维流形的导出辫子理论公理框架，以ribbon张量范畴为系数，通过bar构造控制粘合，证明与普通辫子理论的关系，并给出可计算公式和首次计算。
Miraculous cancellations and the quantum Frobenius for $SL_3$ skein modules
本文在单位根处为任意定向三维流形的 $SL_3$ 绞链模构造了量子 Frobenius 映射，将其推广为 Bonahon-Wong 同态的高秩版本，并借助 Parshall-Wang 的量子群 Frobenius 映射完成构造。
Skein module dimensions of mapping tori of the 2-torus
本文研究了 2- торус 映射射影的 skein 模维数，通过分解扭曲 Hochschild 同调得到通用 q 下的维数计算。
Khovanov skein lasagna modules with $1$-dimensional inputs
本文构造了以 Rozansky-Willis 同调为输入的 Khovanov 辫结拉斯尼亚模量变体，证明其函子性，并给出微分同胚群、Gluck 扭转及 $\mathfrak{gl}_2$ 泡沫的新结果。
The fourth skein module and the Montesinos-Nakanishi conjecture for 3-algebraic links
研究了三维流形的第四skein模的概念，即基于skein关系b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0和框架关系L^{（1）} = aL的skein模，其中a，b_0，b_3可逆.引入了n-代数缠结（和链环）的概念，并分析了3-代数链环的绞链模。作为一个副产品，我们证明了Montesinos-Nakanishi 3-move猜想的3-代数链接（包括3-桥链接）。
We reprove and expand results of Bonahon and Wong on central elements of the Kauffman bracket skein modules at root of 1 and on the existence of the Chebyshev homomorphism, using elementary skein methods.
On the Kauffman bracket skein module of surgery on a (2,2b) torus link
我们证明了在（2，2b）环面链上进行积分手术得到的某些流形的Kauffman括号skein模是可生成的，并列出了一些例子的生成元。
For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links.
Infinitesimal sl(2)-symmetries on the equivariant skein lasagna module
本文构建了sl(2)-作用于等价skein lasagna模块的无穷小对称性，探讨其结构与性质。
We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we …
Askey-Wilson polynomial and genus-two skein module
本文利用亏格二 skein 代数的 q-差分算子，建立了约化 Askey-Wilson 多项式与亏格二环柄体 skein 模之间的对应关系。
Stated Skein Theory and Double Affine Hecke Algebra Representations
本文通过标示skein理论扩展Kauffman括号skein代数，将其嵌入量子6-环面构造洛朗多项式模，从而发展A_1型双重仿射Hecke代数的表示理论，并从纽结补空间构建拓扑定义的模。
This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Markov traces on Hecke algebras to the framework of HOMFLYPT Skein modules. Some open problems on singular Hecke algebras are also presented.
In this paper we disprove a twenty-two year old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid torus. Additionally, we speculate on the structure of the Kauffman bracket skein module of the connected sum of two solid tori.
Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$
本文将 Manolescu-Neithalath 公式解释为同伦余极限，用范畴化投影子计算特定流形的 skein lasagna 模，并证明 $S^2 \times S^2$ 的模平凡，证实 Manolescu 猜想。
In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces $L(p, q)$ via braids.
We introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.
In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order $p$, where $p$ is a prime. The obstruction criteria introduced here extend some results obtained earlier for symmetric spatial graphs.
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