
The BlanchetKhovanov algebras
Abstract Blanchet introduced certain singular cobordisms to fix the functoriality of Khovanov homology. In this paper we introduce graded algebras consisting of such singular cobordisms à la Blanchet. As the main result we give algebraic versions of these algebras using the combinatorics of arc diagrams.A few extra words For an arbitrary field we consider the web algebra . This is a graded algebra which naturally appears as an algebra of singular cobordisms. In particular, it is of topological origin. The underlying category of singular cobordisms was used by Blanchet to fix the functoriality of Khovanov homology. Its objects are certain trivalent graphs and its morphisms are singular cobordisms whose boundary are such trivalent graphs. We call these singular cobordisms foams.An example of such foams is provided by the following relation which holds in the linear category consisting of these singular cobordisms: This relation is called the neck cutting relation and plays a crucial part in the whole story. Working with such foams is topolocially motivated, but hard in practice. In this paper we give an algebraic counterpart of . This provides a direct link between the topological and the algebraic point of view. As a consequence, computations (which are hard to do in practice on the topological side) can be done on the algebraic side, whereas the associativity (a nontrivial fact on the algebraic side) is clear from the topological point of view. 
NEWS
"There are two ways to do mathematics.
The first is to be smarter than everybody else.
The second way is to be stupider than everybody else  but persistent." 
based on a quotation from Raoul Bott.
Upcoming event where you can meet me:
Visit Faro Click

Last update: 20.01.2018 or later ·
email
