Tangling and Braiding the Chessboard Complex

We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Cohen-Macauley. Finally, we discuss a related series of complexes to provide some positive evidence that the braided Houghton groups, introduced by F. Degenhardt, are a series of groups with linearly increasing finiteness length as are the (unbraided) Houghton groups.