This paper studies the (co)homology of rook-Brauer algebras and their subalgebras. Our main results focus on the cohomology of rook-Brauer algebras (which is related to the cohomology of symmetric groups), the cohomology of Motzkin algebras (for which we obtain a vanishing result in positive degrees) and the cohomology of walled Brauer algebras (which is related to the cohomology of products of symmetric groups). Along the way we collect some cohomological analogues of known results for Temperley-Lieb algebras, Brauer algebras and rook algebras.

We would like to thank both James Brotherston and Natasha Cowley for helpful and interesting conversations whilst writing this paper. We would like to thank James Cranch and Sarah Whitehouse for their feedback and support in this project and related work. We'd like to thank Rachael Boyd and Richard Hepworth-Young for interesting conversations at the 2024 British Topology Meeting in Aberdeen. We would like to thank Guy Boyde for his comments and feedback on previous drafts of this paper. We would like to thank the anonymous referee for their helpful comments.