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

TL;DR

This paper introduces a cut-and-paste method for quadriculated disks, explores their tiling parities, and analyzes the arithmetic properties of their adjacency matrices, providing recursive constructions and matrix factorizations.

Contribution

It presents a novel cut-and-paste construction for quadriculated disks, establishes a parity-based tiling matching, and offers a new matrix factorization of the adjacency matrix with integer entries.

Findings

Constructed a recursive cut-and-paste procedure for disks.

Established a parity-based perfect or near-perfect tiling matching.

Factored the adjacency matrix into lower, upper, and modified identity matrices.

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 $\Delta$ 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.