Slab tilings, flips and the triple twist

TL;DR

This paper introduces the triple twist, a new flip invariant for slab tilings, and analyzes its properties, including its range over large regions and connectivity in smaller regions, extending concepts from domino tilings.

Contribution

It defines the triple twist invariant for slab tilings and studies its behavior, providing insights into the structure and connectivity of tilings.

Findings

Triple twist takes many values in large boxes, roughly proportional to the fourth power of volume.

Some smaller regions have tilings connected by flips with a single triple twist value.

The study extends flip invariants from domino to slab tilings, revealing new structural properties.

Abstract

A \textit{domino} is a $2\times 1\times 1$ parallelepiped formed by the union of two unit cubes and a \textit{slab} is a $2\times 2\times 1$ parallelepiped formed by the union of four unit cubes. We are interested in tiling regions formed by the finite union of unit cubes. Domino tilings have been studied before; here we investigate \textit{slab tilings}. As for domino tilings, a flip in a slab tiling is a local move: two neighboring parallel slabs are removed and placed back in a different position. Inspired by the twist for domino tilings, we construct a flip invariant for slab tilings: the \textit{triple twist}, assuming values in $\mathbb{Z}^3$. We show that if the region is a large box then the triple twist assumes a large number of possible values, roughly proportional to the fourth power of the volume. We also give examples of smaller regions for which the set of tilings is connected under flips, so that the triple twist assumes only one value.