Relative Helicity and Tiling Twist

TL;DR

This paper links combinatorial invariants of 3D domino tilings to topological properties of divergence-free vector fields, providing a new geometric interpretation of flux and twist.

Contribution

It introduces a construction associating a divergence-free vector field to domino tilings, interpreting flux as rotation class and twist as helicity.

Findings

Flux corresponds to the rotation class of the vector field.

Twist equals the relative helicity of the vector field.

Provides a topological perspective on tiling invariants.

Abstract

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field $\xi_t$ to any domino tiling $t$, such that the flux of the tiling $t$ can be interpreted as the (relative) rotation class of the field $\xi_t$, while the twist of $t$ is proved to be the relative helicity of the field $\xi_t$.