Flip invariance for domino tilings of three-dimensional regions with two floors

TL;DR

This paper explores the structure of domino tilings in three-dimensional two-floor regions, introducing an algebraic invariant and a new local move called the trit to understand flip connectivity.

Contribution

It introduces an algebraic invariant to characterize flip connected components and a new local move, the trit, enhancing understanding of domino tilings in 3D regions.

Findings

Algebraic invariant nearly characterizes flip components

The trit move connects tilings when floors are identical

Enhanced understanding of flip connectivity in 3D tilings

Abstract

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.