Dissertação de Mestrado - Raphael de Marreiros C. Machado

Domino tilings of 3D cylinders and regularity of disks

In this dissertation we study domino tilings of three-dimensional regions. In particular, we consider the flip connectivity problem for cylinders, i.e., regions of the form D x [0,N]. A flip is a local move: two adjacent dominoes are removed and placed back in a different position. In two dimensions, two domino tilings of the same contractible region are connected by flips. In dimension 3, the problem is subtler. We present the twist, a flip invariant that associates a tiling with an integer number. For many 3D regions, there exist examples of tilings with the same twist which can not be joined by a sequence of flips. Recent papers prove that for certain disks D, called regular, two tilings of the cylinder D x [0,N] with the same twist can almost always be joined by a sequence of flips. Equivalently, a disk D is regular if two tilings of a cylinder of basis D with the same twist can be joined by flips provided that some extra vertical space is allowed. These results are presented and discussed. We then prove regularity or irregularity for new families of quadriculated disks. It turns out that a bottleneck often implies irregularity.