Even Spaced Defects

The homology classes for a unit cell with one or more defects can be found in a sim- ilar way as for the unit cell with no defects. The difference of domino configurations defines the oriented transition graph between them and is a 1-cycle of the chain complex. This 1-cycle is the boundary of a 2-chain - integer combination of the pla- quettes of the underlying square grid - exactly when the two domino configurations are related by a sequence of domino flip moves.

First, we describe the homology group for the unit cell with both side lengths even and containing two fixed defects. This space retracts to the CW-complex shown in Figure 8.1, from which its first homology group is easily deduced to beZ3.

To provide a method for determining the homology class of a particular domino configuration, we introduce a system of cuts as before and measure flows across them. Each cut must assign all boundary elements to the trivial homology class. The choice of cuts to use is related to a choice of representative cycles that provide a basis for the homology. The elements b, -a, and c-a-b can be used for

(0,1,0)

(0,0,1) (1,0,0)

Figure 8.2: For the 12×6 unit cell with 2 defects of size 2×2, the three cuts are shown as red dashed arrows, giving valuesS1,S2, andS3. Each unit cell shows one

of the cycles that together generate representative cycles in each of the homology classes, as black lines with arrows. The sector (S1, S2, S3) is displayed above each

unit cell.

the representative cycles, see Figure 8.1, and cuts are chosen to assign these to the appropriate homology classes, as shown in Figure 8.2, along with the 3 representative cycles mentioned previously. The direction of each cut is shown explicitly, and the flow across the cuts is counted positively where the oriented transition graph passes through the cut from right to left. The homology class is given by a list of the values of the cuts (S1, S2, S3), which we call the sector, analogous to the sector used to

describe the unit cell without defects.

The first two cutsS1andS2are the same as for those of the unit cell without

defects. As was explained in Subsection 2.3.4, a pair of domino configurations with different values for these cuts are not connected by local rearrangements, even when considering them as tiling the plane to become planar domino patterns. This remains true for planar patterns with fixed defects, so we denote the values of these cuts as global homology numbers. The third cutS3 is a different kind of homology

constraint. Figure 8.3 shows two example domino configurations that have different values of S3. Despite the configurations being disconnected in terms of domino

flip moves, there is a finite region around one defect that could be rearranged to transform one configuration into the other. This local rearrangement around the

defect changes the value ofS3; therefore, we call it a defect homology number.

The defect homology number is a weaker constraint than the global homology numbers because the former can be changed by a rearrangement of a finite number of dominoes. However, if the free energy of the molecular system strongly prefers molecules to be adsorbed to the substrate, the desorption-adsorption events would involve as few molecules as possible, which for domino shaped molecules is two. In this case, only domino flip moves are possible and the defect homology numbers are conserved. Otherwise, if several molecules are free to desorb at once, only the global homology numbers are important, and the connectedness of configurations is similar to that of the unit cell without defects. Interestingly, although the homology calculation does not depend on the size or shape of the defects, changing the defect homology number of larger defects would require desorption-adsorption events with a greater number of molecules, meaning larger defects give a stronger constraint.

In the general situation where there are n ≥ 1 defects in a unit cell with even side length, the homology group is Zn+1. To determine the homology class of a particular domino configuration we use n−1 cuts between the defects and 2 around the unit cell. Considering the connectivity network of domino patterns linked by domino flips, sets of domino patterns with larger periodicity are broken up into many more homology classes, compared to the lower periodicity patterns.

The unit cell with even side length and only 1 defect is an exceptional case. In this case, the homology group is Z2, where both the cuts correspond to global homology numbers. This is the same as for the unit cell with no defect. Figure 8.4 gives an example of two domino configurations with an oriented transition graph that wraps around the defect. Although this seems to create a non-trivial cycle, it is possible to expand the cycle away from the defect using domino flips, eventually shrinking it to nothing. The domino configurations generated by this unit cell are those with periodicity equal to the spacing between defects. Therefore, the lowest periodicity domino patterns are unhindered by any homology constraint caused by defects.