Hard squares on cylinders revisited

We consider the independence complexes of square grids with cylindrical boundary conditions. When one of the dimensions is small we use simple reductions induced by edge removals to show explicit natural homotopy equivalences between those spaces. In the second part we expand the results of Jonsson, who calculated the Euler characteristic of cylinders with odd circumference. We describe a series of results for cylinders of even circumference. Finally we define a completely independent combinatorial model (necklaces) which calculates the generating functions of the Euler characteristic of cylindrical grids. We conjecture that this model has some particularly simple structure.