通过网和泡沫的泛SL（2）上同调

组合与规范理论的泡沫求值函子并不相同

The combinatorial and gauge-theoretic foam evaluation functors are not the same

本文通过构造反例证明 Kronheimer-Mrowka 的规范理论网函子 J^sharp 在平面网子范畴上的限制与 Khovanov-Robert 证明良定义的组合替代函子 J^flat 并不等价，澄清了二者关系。

缠结（n > 3）的泡沫与sl（n）上同调

Foams and sl(n) cohomology for tangles (n > 3)

我们使用4-valent可平面图和奇异协边（称为foams）来构造缠结的积分双阶上同调，特别是对于链环，其阶化欧拉示性数产生sl（n）链环多项式（n > 3）.

Computer Bounds for Kronheimer-Mrowka Foam Evaluation

Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor $J^\sharp$, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement $J^\flat$ for $J^\sharp$. Of particular interest is the relationship between the dimension of $J^\flat(K)$ for a web $K$ and the number of Tait colorings $\mathrm{Tait}(K)$ of $K$; these two numbers are k…

泛Khovanov-Rozansky sl（2）上同调

Universal Khovanov-Rozansky sl(2) cohomology

本文利用依赖于两个参数的齐次势推广了n=2的Khovanov-Rozansky上同调，得到了泛Khovanov-Rozansky sl（2）链上同调.这个理论等价于泛泡沫sl（2）链上同调，在用适当的环张紧两个理论之后。

A closed formula for the evaluation of $\mathfrak{sl}_N$-foams

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag manifolds.

Symmetric Khovanov--Rozansky link homologies

We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

色缠结的上同调理论

A cohomology theory for colored tangles

本文利用sl（2）泡沫上同调定义了一个关于定向框架缠结的上同调理论，其分量标为U_q（sl（2））的不可约表示.我们表明，相应的有色不变量的缠结可以组装成更大的缠结的不变量。对于节点和链接的情况下，相应的理论是一个分类的有色琼斯多项式，并提供了一个工具，有效地计算所得的有色不变量的节点和链接。我们的理论定义在高斯整数Z[i]上（更一般地，定义在Z[i][a，h]上，其中a，h是形式参数），并增强了有色琼斯多项式的现有简化。

函子量子sl（k）纽结不变量的组合方法

A combinatorial approach to functorial quantum sl(k) knot invariants

本文利用O范畴的抛物奇异块对sl（k）链不变量进行了分类。我们的方法旨在尽可能初级，为Sussan的主要结果提供组合证明。本文首先构造了一个满足MOY关系的网或“特殊”三价图（标号为1，2，k-1，k）的正合函子值不变量。然后，我们通过传递到导出范畴，将其扩展到链接的sl（k）-不变量。在这种情况下，使用泡沫的方法自然出现。更一般地说，我们希望我们的方法提供了一个表示理论解释的sl（k）-同调，泡沫和Kapustin-Lie公式的基础上。这意味着Khovanov-Rozansky链的同调是通过限制从我们的不变量中得到的。

Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we …

环带上的SL(2)与SL(3)编织代数

Annular SL(2) and SL(3) web algebras

本文利用环带泡沫拓扑量子场论定义环带上的等变SL(2)与SL(3)编织代数，并建立缠结不变量及非椭圆编织与权格闭合路径的双射对应。