**Data**

- Title: Generic -foams, web and arc algebras
- Authors: Michael Ehrig, Catharina Stroppel and Daniel Tubbenhauer
- Status: Preprint. Last update: Thu, 22 Oct 2020 14:24:54 UTC
- ArXiv link: http://arxiv.org/abs/1601.08010
- LaTex Beamer presentation: Slides1, Slides2

**Abstract**

We define parameter dependent -foams and
their associated web and arc algebras and verify that they specialize to several
known or
constructions related to higher link and tangle
invariants.
Moreover, we show that all these specializations are equivalent, and
we deduce several applications, e.g. for the associated link
and tangle invariants, and their functoriality.

**A few extra words**

Let
be a (certain) set of generic parameters.
In this paper we introduce
a -version of singular topological quantum field theories (TQFTs)
which we use to define a certain -parameter
foam -category
, that is a
certain -category of topological origin.
We obtain from several specializations.
Among the specializations of this parameter
version one can find the main foam -categories
studied in the context
of higher link and tangle invariants:

- Khovanov/Bar-Natan's cobordisms
- Caprau's “foams”
- Clark-Morrison-Walker's disoriented cobordisms
- Blanchet's foams

A relation that holds for these parameter foams is the following:

We also study the
web algebra
corresponding to , i.e. an algebra
which has an associated -category of certain bimodules
giving a (fully) faithful -representation
of .
Similarly, we study also an algebraic model called arc algebra.

Our main result is, surprisingly,
that any two specializations (under some conditions) are
isomorphic/equivalent
(our results are even stronger since everything is explicit).
Similarly for the associated -categories
of bimodules and foams.
This includes the four examples from above.

We discuss several applications of our explicit isomorphisms/equivalences.
For example, we can show the
higher tangle invariants constructed from the various
-categories
are the same (they get identified by the above
equivalence)
and not just the associated link homologies.