Relative Helicity and Tiling Twist

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 $\hat{I}{¾}_t$ to any domino tiling $t$, such that the flux of the tiling $t$ can be interpreted as the (relative) rotation class of the field $\hat{I}{¾}_t$, while the twist of $t$ is proved to be the relative helicity of the field $\hat{I}{¾}_t$.

21 pages, 19 figures. Two figures and minor clarifications added from previous version. To appear in the Transactions of the American Mathematical Society