Some Polycubes Have No Edge Zipper Unfolding

It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no edge zipper unfolding where the cut edges are further restricted to form a path.