<p>When knotted or linked vortex tubes are considered in real-analytic steady Euler flows, the flows should be Beltrami flows with constant proportionality factors that have chaotic streamlines. In this study, four types of such Beltrami flows were derived on the assumption that the set of streamlines in each flow had hexagonal symmetry. Their systematic derivation was enabled by information provided via crystallography, which is applicable to spatially periodic objects not restricted to chemical materials. Invariant tori, which are stream and vortex tubes in Beltrami flows, were numerically investigated using various proportionality factors and initial points for the derived hexagonal flows. As a result, a variety of knotted or linked invariant tori were found to be arranged as atoms in hexagonal crystals. Some invariant tori were observed to form infinitely spreading chains with link structures similar to those of chain-mail-like polycatenanes in chemistry.</p>