Frobenius Modules and Essential Surface Cobordisms

Generalized Skein Modules of Surfaces

Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are relevant to generalized sl(n) link homology theories. We also investigate the skein modules that result from embedding these marked cobordisms within 3-manifolds, and arrive at an explicit presentation for several of these generalized skein modules.

关于可定向二维带状体的TQFT注记

A note on TQFTs for orientable 2-dimensional cobordisms

本文提出一个中间框架，用于分类可定向二维带状体的TQFT，以解决非定向情况下分类所需的额外数据问题。

On Link Homology Theories from Extended Cobordisms

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webs…

Twisted homology cobordism invariants of knots in aspherical manifolds

We fix a null-homologous, homotopically essential knot $J$ in a 3-manifold with PTFA fundamental group and study concordance of knots that are homotopic to $J$. We construct an infinite family of knots that are characteristic to $J$, and then use $L^2$-methods to show that they are not concordant to $J$.

一个表面 skein TQFT 的构造及其扩展到 4 维 2-柄体

A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies

本文提出了一种构造表面 skein TQFT 的方法，并扩展其到 4 维 2-柄体，通过归纳状态和构造和 Kirby 颜色来计算不变量。

Bar-Natan theory and tunneling between incompressible surfaces in 3-manifolds

The author defined for each (commutative) Frobenius algebra a skein module of surfaces in a $3$-manifold $M$ bounding a closed $1$-manifold $α\subset \partial M$. The surface components are colored by elements of the Frobenius algebra. The modules are called the Bar-Natan modules of $(M,α)$. In this article we show that Bar-Natan modules are colimit modules of functors associated to Frobenius algebras, decoupling topology from algebra. The functors are defined on a category of $3$-dimensional compression bordisms embedded in cylinders over $M$ and take values in a linear category defined from …