This is part of the additional material for the preprint Domino tilings of three-dimensional regions: flips, trits and twists.
Below are some details about the flip connected components of the 4x4x4 box. This box has 5051532105 tilings, divided into 93 connected components, which we number 0 to 92 (the order is arbitrary).
Below is a table with number of elements and twist of each flip connected component:
| Connected component | Number of tilings | Twist |
|---|---|---|
| 0 | 4412646453 | 0 |
| 1 | 310185960 | 1 |
| 2 | 310185960 | -1 |
| 8 | 8237514 | 2 |
| 7 | 8237514 | -2 |
| 3 | 718308 | 2 |
| 5 | 718308 | -2 |
| 4, 6 | 283044 | 0 |
| 27, 36, 58 | 2576 | 3 |
| 28, 35, 57 | 2576 | -3 |
| 9, 13, 22, 23, 29, 32, 37, 43, 49, 55, 76, 80 | 618 | 3 |
| 12, 16, 21, 25, 30, 31, 39, 45, 51, 56, 75, 77 | 618 | -3 |
| 10, 15, 24, 34, 38, 42, 47, 52, 54, 59, 78, 82 | 236 | 1 |
| 11, 14, 26, 33, 40, 41, 44, 50, 53, 60, 79, 81 | 236 | -1 |
| 48, 61, 83 | 4 | 4 |
| 46, 62, 84 | 4 | -4 |
| 17, 63, 67, 72, 85, 92 | 1 | 4 |
| 18, 19, 64, 65, 68, 70, 71, 73, 86, 87, 90, 91 | 1 | 0 |
| 20, 66, 69, 74, 88, 89 | 1 | -4 |