Skein Modules and Lattice Gauge Field Theory

The $\mathfrak{gl}_2$-Skein Module of Lens Spaces via the Torus and Solid Torus

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

We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.

On the Kauffman bracket skein module of the quaternionic manifold

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, we determine that they are linearly independent.

Frobenius algebras and skein modules of surfaces in 3-manifolds

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing…

Deformation of string topology into homotopy skein modules

Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces.

On the skein module of the product of a surface and a circle

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

Let k be an integral domain containing the invertible elements α, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein module of handlebodies. Gilmer and Zhong have studied the Homflypt skein modules of a connected sum of two 3-manifolds, here we study the case for the Kauffman skein module and show that K(M_1 # M_2) is isomorphic to K(M_1) tensor K(M_2) over a certain localized ring, where M_1 # M_…

Skein lasagna modules for 2-handlebodies

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 $S^2$.

The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds

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…

$C$型skein模与透明元素

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

Let k be a subring of the field of rational functions in α, s which contains α^{1}, α^{-1}, s^{1}, s^{-1}, . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the free k-module generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of k={Q}(α, s), the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S^{1} \times S^2)…

Skein lasagna 模与环柄分解

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。

Skein Modules from Skew Howe Duality and Affine Extensions

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…

A basis for the Kauffman skein module of the product of a surface and a circle

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 $2^{2g+1} + 2g -1$.

stated skein代数的有限表示与格点规范场论

Finite presentations for stated skein algebras and lattice gauge field theory

该研究给出stated skein代数的有限表示，证明其为Koszul代数且与格点规范场论的量子模代数同构，推广了前人相关研究结果。

Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main …

四维 skein 模块、处理附件与结

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

For each closed, orientable surface F, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module K_t(F x [0,1]). The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = - 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of F.

The Homflypt skein module of a connected sum of 3-manifolds

If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the similar result holds for relative skein modules. If M contains a separating 2-sphere, we give conditions under which certain relative skein modules of M vanish over specified localized rings.

On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism

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…

The Kauffman bracket skein module of two-bridge links

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.

不可约表示下的 Kauffman 括号 skein 模

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

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology. We prove the basic properties of our homology. In particular, the short exact sequence of homologies of skein related tangles and the Kunneth formula for the tensor product of tangles.

闭3-流形的绞纱模在特征簇上定义线丛

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

We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein module of a cylinder over the torus.

Skein modules

We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one should start from knots in 3-manifolds, surfaces in 4-manifolds, etc. However our theory is, until now, developed only in the case of links in 3-manifolds, with only a glance towards 4-manifolds. The main object of the theory is a skein module and we devote this chapter mostly to the description of skein modules in 3-dimensio…

辫子视角下的 skein 模综述

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。

3-流形的带标记纽结模与拓扑量子场论

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。

The finiteness conjecture for skein modules

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

The Kauffman bracket skein modules, S(M,A), have been calculated for A=+1,-1, for all 3-manifolds M by relating them to the SL(2,C)-character varieties. We extend this description to the case when A is a 4-th root of 1 and M is either a surface x [0,1] or a rational homology sphere (or its submanifold).

HOMFLYPT Skein Theory, String Topology and 2-Categories

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

We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other component is limited to the number of twists.

Steinberg skein identities at roots of unity

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.

辫理论、线缺陷与量子对称对

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。

关于 $(S^1 \times S^2) \\ \\# \\ (S^1 \times S^2)$ 的 Kauffman 括号 skein 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

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is computed.

Kauffman bracket versus Jones polynomial skein modules

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.

导出辫子模

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。

$SL_3$ 绞链模中的奇迹相消与量子 Frobenius

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。

2- торус映射射影的 skein 模维数

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 辫结拉斯尼亚模量，输入为 $1$ 维

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

We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-move conjecture for 3-algebraic links (including 3-bridge links).

On Kauffman Bracket Skein Modules at Root of Unity

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

We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples.

HOMFLY-PT skein module of singular links in the three-sphere

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.

无穷小sl(2)-对称性在等价 skein lasagna 模块上

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。

Quantum torus methods for Kauffman bracket skein modules

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 多项式与亏格二 skein 模

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。

标示Skein理论与双重仿射Hecke代数表示

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。

HOMFLYPT Skein module of singular links

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.

Kauffman Bracket Skein Module of the Connected Sum of Handlebodies: A Counterexample

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 带、范畴化投影子与 $S^{2}\times{S^{2}}$ 的 skein lasagna 模

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。

An alternative basis for the Kauffman bracket skein module of the Solid Torus via braids

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.

The Torelli group and the Kauffman bracket skein module

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.