An invariant of tangle cobordisms via subquotients of arc rings

Khovanov's homology for tangles and cobordisms

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a `TQFT') to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

An invariant of tangle cobordisms

We prove that the construction of our previous paper math.QA/0103190 yields an invariant of tangle cobordisms.

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

We use 4-valent planar graphs and singular cobordisms (called foams) to construct an integral doubly-graded cohomology for tangles, and in particular for links, whose graded Euler characteristic yields the sl(n) link polynomial (for n > 3).

从有限状态自动机到缠结配边：一场从一维到四维的TQFT之旅

From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions

本文概述了范畴化Reshetikhin--Turaev $\mathsf{SL}(N)$-量子链环不变量的链环同调理论，并以有限状态自动机与一维布尔TQFT的联系作为引入。

A cobordism category attached to Khovanov-Rozansky link homologies based on operads

We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist th…

关于sl(2)链环同调的函子性

On the functoriality of sl(2) tangle homology

本研究构建gl(2)网与泡沫范畴和Bar-Natan的Temperley-Lieb图配边范畴的显式等价，定义弧代数的网版本及拟遗传覆盖以得到严格函子性的链环缠结同调，还给出其与原代数的同构，实现环形链环同调量子化的严格函子性版本。