Abstract
We consider the problems of unfolding lattice polygons lying in a width-2 lattice of unit height. During the unfolding process, all linkage edges are preserve and no edge crossing are allowed. Let n be the depth of edges of the given lattice polygon. We first show that a unknotted lattice polygon lying in a 3D width-2 lattice of unit height can be unfolded in O(n^2) moves and time. We then show that a unknotted lattice polygon lying in a 3D 3x3 lattice of unit height can be reconfigured to a 2x2 lattice polygon. The main technique in our algorithms is to fold up all blocks of the lattice polygon from the rightmost cubic cells of the given lattice. We hope that our results shed some light on solving the more general conjectures, which we proposed, that a 3D unknotted lattice polygon lying in any lattice can always be unfolded.