Periodic entanglement I: networks from hyperbolic reticulations

High-symmetry free tilings of the two-dimensional hyperbolic plane ({ H}^2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.