Skein modules at the 4th roots of unity

The Kauffman skein algebra of a surface at $\sqrt{-1}$

We study the structure of the Kauffman algebra of a surface with parameter equal to sqrt(-1). We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat connections in a trivial SU(2)-bundle over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity parametrizing the TQFT accumulates to a root of unity. We interpret the case of sqrt(-1) in terms of parallel transport operators.

单位根处陈述 skein 表示之经典阴影

Classical shadows of stated skein representations at roots of unity

本文将闭曲面 skein 代数相关结果推广至开曲面的陈述 skein 代数，证明 +1 参数代数嵌入奇次单位根参数代数的中心，并构造其与推广 SL2 特征簇正则函数代数的同构，从而为不可约或局部表示关联特征簇中的不变量。

有标记 skein 代数的中心

Center of the stated skein algebra

本文研究单位根情形下有标记 skein 代数的中心结构，给出其简单描述并计算该代数作为中心上模的维数。

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。

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.

$C$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。

源自Kauffman括号skein代数的Frobenius代数

Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra

本文研究了特定曲面在$2N$次单位根处的skein模，并由此构造出Frobenius代数。

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。

Dimension and Trace of the Kauffman Bracket Skein Algebra

Let $F$ be a finite type surface and $ζ$ a complex root of unity. The Kauffman bracket skein algebra $K_ζ(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_ζ(F)$ over its center, and we extend a theorem of Frohman and Kania-Bartoszynska which says the skein algebra has a splitting coming from two pants decompositions of $F$.

切片 skein 代数与几何 Kauffman 括号

Sliced skein algebras and geometric Kauffman bracket

本文研究有限型曲面的切片 skein 代数，证明其为整环，计算单位根时的中心与 PI 次数，并引入 ρ-约化 skein 模，证明其维数性质与单模性。

分级球面 skein 2+1-G-HQFT 及修改后的 Turaev-Viro 不变量

Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants

本文提出基于 G 分级的染色映射和 skein 模块，用于定义 2+1-G-HQFT，并恢复修改后的 Turaev-Viro 三维不变量。

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…

$SL_3$ 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。

The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arXiv:1206.1638).

The Localized Skein Algebra is Frobenius

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters $χ(F)$ of the fundamental group of $F$. We localize by inverting the nonzero characters to get an algebra $S^{-1}K_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_N(F)$ is a finite rank module over $χ(F…

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。

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。

Faithfullness of geometric action of skein algebras

We show that the action of the Kauffman bracket skein algebra of a surface $Σ$ on the skein module of the handlebody bounded by $Σ$ is faithful if and only if the quantum parameter is not a root of 1.

Stated skein modules of marked 3-manifolds/surfaces, a survey

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces), describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representati…

广义 skein 代数的中心

Center of generalized skein algebras

本文研究了广义 skein 代数的中心结构，揭示了其在量子参数条件下几乎 Azumaya 的性质，并探讨了其在表示论中的影响。

从 Dehn-Thurston 坐标看曲面辫代数到量子环面的嵌入

An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

本文利用辫代数在柄体辫模上的作用，构造了曲面 Kauffman 括号辫代数到局部化量子环面的嵌入，并借此研究了单位根处的表示，给出了 Bonahon-Wong 唯一性猜想的新证明及唯一表示的显式重构方法。

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 …

Traces on the skein algebra of the torus

For a surface $F$, the Kauffman bracket skein module of $F \times [0,1]$, denoted $K(F)$, admits a natural multiplication which makes it an algebra. When specialized at a complex number $t$, nonzero and not a root of unity, we have $K_t(F)$, a vector space over $\mathbb{C}$. In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space $K_t(T^2)$ has five distinct traces. One trace, the Yang-Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on $K_t(T^2)$ correspond to each of the four $\mathbb{Z}_2$ ho…

$\rm{PGL}_2$ 在单位根处的蛛网

$\rm{PGL}_2$ webs at roots of unity

本文研究 $\rm{PGL}_2$ skein 理论在单位根处的蛛网，构造切比雪夫多项式以得到中心元素，并证明偶数阶单位根时环面 Frobenius 同态非单射。

Evaluable Jones-Wenzl idempotents at root of unity and modular representation on the center of U_qsl(2)

Let p an integer. We define a family of idempotents (and nilpotents) in the Temperley - Lieb algebras at 4p-th roots of unity which generalize the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group Uqsl(2), where q is a 2p-th root of unity. In the manner of the [BHMV95] topological quantum field theorie (TQFT), they provide a canonical basis in colored skein modules to define mapping class groups representations. In the torus case, this basis allows us to obtain a partial mat…

On invariants of graphs related to quantum $\mathfrak{sl}(2)$ at roots of unity

We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

Topological invariants from quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd order. These constructions use the notion of modified trace and relative $\mathit{G}$-modular category.

Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group of S to SL_2(C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental rol…

Gilmer-Masbaum 映射在 Kauffman 括号 skein 模上不是单射的

The Gilmer-Masbaum map is not injective on the Kauffman bracket skein module

本文通过计算 2-环面映射环面族的求值映射像，发现 Gilmer-Masbaum 映射在某些同调类上非单射，回答了其维数下界问题，并给出了 Dehn 扭转幂映射环面的 Kauffman 括号 skein 模的基。

非半单量子不变量与阿贝尔经典阴影

Non-semisimple quantum invariants and abelian classical shadows

本研究借助奇数次单位根下3-流形的$U_q^Hsl_2$非半单不变量，在对应考夫曼括号链环模上构造出经典阴影可为任意阿贝尔非中心特征的映射。

A categorification of the colored Jones polynomial at a root of unity

There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.

Diagram categories for $\textbf{U}_q$-tilting modules at roots of unity

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.

Overview of knot invariants at roots of unity

We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to $\mathcal{U}_q(sl_N)$ and highlight the connections between different invariants.

All roots of unity are detected by the A-polynomial

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.

Examples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points whose associated roots of unity are not 1 or -1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than 1 and -1 occur.

Infinitely many roots of unity are zeros of some Jones polynomials

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.

Dehn填充的 skein 模的维数

Dimension of the skein module of a Dehn filling

研究Dehn填充的skein模的维数，基于两桥结的条件，证明了在特定斜率和单位根下，skein模的维数与生成元有关。

稳定Khovanov同调与体积

Stable Khovanov homology and Volume

本文证明高度扭转链环的n着色Jones多项式在直接极限下趋近于skein元Kauffman括号的范畴化，证实了Thurston双曲Dehn手术定理的量子版本，并基于体积猜想计算了其渐近增长率与正则理想八面体体积的关系。