  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 $\xi_t$.
