Detecting torsion in skein modules using Hochschild homology

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。

The Kauffman skein module at first order

For a 3-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+ε$ where $ε^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$ which we prove in particular cases. As a maybe useful tool, we then introduce a notion of twisted self-linking and prove that it satisfies the Kauffman relations at first order. These questions come from considerations on asymptotics of quantum invariants.

Unwinding the relative Tate diagonal

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This allows us to show that the target of this spectral sequence can be identified with Hochschild homology when the topological Hochschild homology is torsion-free as a module over $\mathrm{THH}_*(\mathbb{F}_2)$, parallel to results of Mathew on degeneration of the Hodge-to-de Rham spectral sequence. To carry this out, we apply work of Nikolaus-Scholze to develop a general Tate diagonal for Hochschild…

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。

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 The Homflypt Skein Module of S^1 x S^2

Let $k$ be a subring of the field of rational functions in $x, v, s$ which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$. If $M$ is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of $M$ over $k$. This is the free $k$-module generated by isotopy classes of framed oriented links in $M$ quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$; (2) $L$ with a positive twist $=(xv^{-1})L$; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where $O$ is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the…

Khovanov Homology: categorification of the Kauffman bracket relation

Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a simple description of Khovanov long exact sequence of homology. Khovanov homology, over the field Q, is a categorification of the Jon…

三流形在单位根处的skein代数

Skein Algebras of Three-Manifolds at 4th Roots of Unity

本文在任意3-流形的skein模中引入代数结构，证明当参数为±i且H_1(M;Z)无2-挠时该代数与参数为±1时同构，并揭示其为PSL_2(C)-特征簇的非简化坐标环。

通过辫子方法计算 $S^1 \times S^2$ 的 HOMFLYPT 绞线模

The HOMFLYPT skein module of $S^1 \times S^2$ via braids

本文通过辫子理论技术，将实心环面中的 Lambropoulou 不变量推广到 $S^1 \times S^2$，并求解无穷方程组，从而完全确定其 HOMFLYPT 绞线模的结构：自由部分仅由空链环生成，其余元素均为挠元。

基于辫子的 $S^1\times S^2$ 的考夫曼括号扭结模

The Kauffman bracket skein module of $S^1\times S^2$ via braids

该研究提出两种基于辫子计算 $S^1\times S^2$ 的考夫曼括号扭结模的方法，证明其非无挠且自由部由平凡纽结生成，还通过图解法得到其挠部的闭式公式。

Torsion in Graph Homology

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$, we determine precisely those graphs whose cohomology contains torsion. For a larger class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. The ideas of this paper could potentially be used to predict the Khovanov-Rozansky $sl(m)$ homology of knots (in particular $(2,n)$ torus knots). We also …

半充分链环的Khovanov同调中的挠

Torsion in Khovanov homology of semi-adequate links

针对Khovanov链环同调中$\text{Z}_2$-挠存在性的Shumakovitch猜想，研究者借助图的色上同调分析半充分链环的Khovanov同调挠，得到显式公式并证明对应Tait型图有长度≥3的环时存在$\text{Z}_2$-挠，还提出挠与辫指数相关的猜想。

Homology TQFT's and the Alexander-Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find…

On the first group of the chromatic cohomology of graphs

The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homology of A_m and the only torsion, equal to Z_m, appears in gradings (i,m(i+1)/2) for any positive odd i. We analyze here the grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1,v-1}_{A_2}(G) and H^{1,2v-3}_{A_3}(G). The grou…

有理同调3-球面与SL(2,C)表示

Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations

研究有理同调3-球面的SL(2,C)不可约表示，证明当3-流形的基环非平凡且为2- torsion或3- torsion时，存在不可约表示，并应用此结果探讨Kaufmann括号 skein 模块的 torsion 性质。