Configuration spaces of squares in a rectangle

Abstract

The configuration space $F_k\left(Q,r\right)$ 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 $Q^k$. The critical points of the function $𝜃$ are described in geometric and combinatorial terms. We also show that under certain conditions, the space $F_k\left(Q,r\right)$ is connected.