DOMINO TILINGS OF 3D CYLINDERS AND REGULARITY OF DISKS
RAPHAEL DE MARREIROS C MACHADO · 2021
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 ×[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 an integer number with a tiling.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 × [0, N ] with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder.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.