Domino tilings of three-dimensional regions

TL;DR

This thesis explores the structure and invariants of domino tilings in three-dimensional regions, introducing new polynomial and integer invariants, and analyzing their properties and connectivity via local moves.

Contribution

It introduces a polynomial invariant and a twist invariant for 3D domino tilings, providing new tools to understand tiling connectivity and structure.

Findings

The space of tilings of regions of the form D×[0,2] is connected by flips and trits.

A polynomial invariant P_t(q) characterizes tilings nearly in the same connected component.

The twist invariant has a simple combinatorial formula and relates to knot theory.

Abstract

In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form $\mathcal{D} \times [0,N]$. In particular, we investigate the connected components of the space of tilings of such regions by flips, the local move performed by removing two adjacent dominoes and placing them back in the only other possible position. For regions of the form $\mathcal{D} \times [0,2]$, we define a polynomial invariant $P_t(q)$ that characterizes tilings that are "almost in the same connected component", in a sense discussed in the thesis. We also prove that the space of domino tilings of such a region is connected by flips and trits, a local move performed by removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position. For the general case, the invariant is an integer, the twist, to which we give a simple combinatorial formula and an interpretation via knot theory; we also prove that the twist has additive properties for suitable decompositions of a region. Finally, we investigate the range of possible values for the twist of tilings of an $L \times M \times N$ box.