  In this paper we study handlebody versions of some classical diagram
algebras, most prominently, handlebody versions of Temperley-Lieb, blob,
Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by
Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of
(sandwich, inflated or affine) cellular algebras. We explain this reformulation
and how all of the above algebras are part of this theory.
