Arithmetic properties of the adjacency matrix of quadriculated disks

Abstract

Let $Δ$ be a bicolored quadriculated disk with black-to-white matrix $B_Δ$. We show how to factor $B_Δ= L\tilde DU$, 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.