  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.
