Configuration spaces of squares in a rectangle

Abstract

The configuration space Fk(Q,r) of k squares of size r in a rectangle Q is studied with the help of the tautological function 𝜃 defined on the affine polytope complex Qk. The critical points of the function 𝜃 are described in geometric and combinatorial terms. We also show that under certain conditions, the space Fk(Q,r) is connected.