Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk $\mathrm{\Delta }$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_{\mathrm{\Delta }}$ be the black-to-white adjacency matrix: we factor $B_{\mathrm{\Delta }}=L\tilde{D}U$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde{D}$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde{D}$ and $U$ are equal to 0, 1 or -1.