Binomial ideals of domino tilings

TL;DR

This paper explores the algebraic structure of domino tilings of cubiculated regions, connecting tiling spaces with algebraic ideals and providing new insights into their flip connectivity using toric algebra methods.

Contribution

It introduces tiling and flip ideals, describes moves connecting tilings via binomials, and proves quadratic binomial generation for 2D simply connected regions.

Findings

Describes moves connecting tilings using algebraic binomials

Introduces tiling and flip ideals to formalize flip connectivity

Shows all binomials in 2D simply connected regions are quadratic

Abstract

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if $R$ is a $2$-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of $R$ can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a $2$-dimensional simply connected region is connected by flips.