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

Abstract: 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 receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let be the black-to-white adjacency matrix: we factor , where and are lower and upper triangular matrices, is obtained from a larger identity matrix by removing rows and columns and all entries of , and are equal to , or . [Copyright &y& Elsevier]