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

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

2009-04-30

arxiv.org/abs/0904.4814v1

Linear Algebra and its Applications 432 (2010) 2423-2437

Mathematics

Combinatorics

20 pages, 17 figures

Scientific paper

10.1016/j.laa.2009.09.003

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 $\Delta$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_\Delta$ be the black-to-white adjacency matrix: we factor $B_\Delta = L\tilde DU$, 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.

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