The graph dual of Plantri 9-12, and its realization as knotology weaving
A deltahedron is a polyhedron where all faces are equilateral triangles. A deltahedron is unicursal if its skeletal graph is a Tait graph of a knot. The "knot" in such case is a kagome basket, the weaving of which is easily encoded because of the simple structure of a knot as opposed to a multicomponent link.
A 3-connected triangulation of the sphere might describe a deltahedron if the geometry works for equilateral faces. The Plantri software which is built in to SageMath, can generate 3-connected triangulations. These can be filtered for unicursality by counting spanning trees (another feature built in to Sage Math): a plane graph is unicursal if it has an odd number of spanning trees. In the list of candidate unicursal deltahedra given here, the results have been further filtered to have no vertex of degree 7 or higher, and to have some symmetry, |Aut|>1, as these are perhaps the most interesting to weave.
The "road codes" given will be explained in a later post. The "Plantri identifiers" given here are the number of vertices in the triangulation followed by the listing ordinal when Plantri is asked to generate all 3-connected triangulations on the sphere with that number of vertices.
Unicursal polyhedral triangulations with |Aut|>1 and vertex degrees < 7
Plantri 5-0: |Aut|= 12 Vdeg = [2 3]
7.2 0.9 5.10 12.3 13.8 1.14 6.15 11.16 17.4
Plantri 7-1: |Aut|= 4 Vdeg = [2 3 0 2]
0.9 11.6 3.14 16.1 8.17 19.10 20.5 13.22 2.23 24.15 25.4 21.26 12.27 7.28 29.18
Plantri 7-2: |Aut|= 6 Vdeg = [3 0 3 1]
0.11 12.5 7.14 16.9 3.18 20.1 10.21 22.15 23.8 17.24 2.25 26.19 27.4 13.28 6.29
Plantri 7-3: |Aut|= 20 Vdeg = [0 5 2]
7.0 3.10 13.6 9.16 2.17 19.12 20.5 15.22 8.23 1.24 25.18 26.11 27.4 21.28 14.29
Plantri 8-8: |Aut|= 2 Vdeg = [1 3 3 1]
11.2 4.13 16.9 17.0 6.19 15.22 23.10 24.1 25.18 7.26 21.28 14.29 30.3 31.12 5.32 33.20 34.27 35.8
Plantri 9-12: |Aut|= 2 Vdeg = [2 2 2 3]
4.11 14.1 17.6 8.19 2.23 24.13 25.0 15.26 28.21 30.9 18.31 7.32 33.20 29.34 35.10 5.36 16.37 27.38 39.22 3.40 41.12
Plantri 9-27: |Aut|= 2 Vdeg = [1 4 1 3]
1.8 11.4 2.17 9.18 21.6 24.15 13.26 23.28 29.16 3.30 10.31 19.32 33.0 34.7 35.22 36.27 37.14 25.38 12.39 5.40 20.41
Plantri 9-31: |Aut|= 4 Vdeg = [2 3 0 4]
12.3 5.14 17.8 11.20 21.4 22.13 2.23 25.0 6.27 15.28 30.9 31.18 24.33 34.1 35.26 7.36 16.37 29.38 39.10 40.19 41.32
Plantri 9-32: |Aut|= 2 Vdeg = [2 1 4 2]
4.11 13.2 16.9 7.18 1.22 12.23 24.3 25.14 20.27 30.5 10.31 32.15 33.26 21.34 0.35 29.36 37.6 38.17 8.39 19.40 41.28
Plantri 9-40: |Aut|= 4 Vdeg = [0 5 2 2]
11.2 6.15 18.3 19.12 0.21 9.22 16.25 26.5 28.13 29.20 1.30 10.31 23.32 34.7 14.35 36.27 4.37 38.17 24.39 33.40 41.8
Plantri 9-47: |Aut|= 2 Vdeg = [1 2 5 1]
9.2 7.14 17.4 19.12 21.0 11.24 18.25 5.26 28.15 29.8 30.1 31.22 20.33 13.34 6.35 27.36 37.16 38.3 39.10 40.23 41.32
Plantri 10-140: |Aut|= 2 Vdeg = [1 4 1 4]
0.9 11.2 5.14 21.4 22.13 7.24 17.26 29.20 30.3 31.12 23.32 6.33 15.34 36.27 37.18 39.10 40.1 8.41 25.42 16.43 35.44 45.28 46.19 47.38
Plantri 10-141: |Aut|= 6 Vdeg = [0 3 6 1]
11.2 7.16 19.10 20.1 13.22 5.24 27.18 28.9 14.31 23.32 4.33 35.26 36.17 37.8 29.38 39.0 21.40 12.41 3.42 43.34 44.25 45.6 46.15 30.47
Plantri 10-176: |Aut|= 2 Vdeg = [1 3 3 3]
8.1 6.15 17.4 21.12 23.0 9.24 18.27 5.28 29.16 30.3 32.25 33.10 22.35 13.36 38.19 26.39 40.31 2.41 42.7 14.43 37.44 45.20 46.11 47.34
Plantri 11-382: |Aut|= 2 Vdeg = [2 2 2 5]
13.0 7.20 22.5 24.17 11.26 28.1 29.14 12.31 27.32 33.2 35.16 25.36 10.37 39.4 23.40 41.18 8.43 21.44 45.6 46.19 47.42 9.48 49.38 50.3 51.34 52.15 53.30
Plantri 11-719: |Aut|= 2 Vdeg = [2 2 2 5]
12.1 7.16 21.6 22.15 9.24 27.4 30.13 0.31 11.32 33.2 28.35 5.36 20.37 39.18 41.26 42.3 43.34 29.44 45.14 23.46 8.47 17.48 38.49 50.19 51.40 52.25 53.10
Plantri 11-724: |Aut|= 2 Vdeg = [0 4 4 3]
3.14 17.6 12.21 0.23 24.9 27.20 13.28 2.29 32.7 33.18 4.35 15.36 31.38 39.8 25.40 11.42 43.22 1.44 45.30 46.37 47.16 48.5 34.49 19.50 26.51 41.52 10.53
Plantri 11-739: |Aut|= 2 Vdeg = [0 5 2 4]
1.10 17.0 18.9 6.21 14.23 26.11 27.2 5.30 31.22 15.32 25.34 35.12 37.4 38.29 40.19 8.41 16.43 33.44 24.45 46.13 47.36 48.3 49.28 39.50 51.20 7.52 53.42
Plantri 11-976: |Aut|= 2 Vdeg = [1 4 1 5]
15.2 8.17 20.5 7.22 23.18 27.14 28.1 11.30 24.33 19.34 35.6 36.21 4.37 39.26 40.13 41.0 29.42 10.43 45.32 25.46 38.47 48.3 49.16 9.50 51.44 52.31 53.12
Plantri 11-980: |Aut|= 4 Vdeg = [2 3 0 6]
9.0 6.15 17.4 2.23 25.8 11.28 30.21 19.32 34.13 27.36 10.37 1.38 39.24 40.7 14.41 42.33 43.20 31.44 18.45 5.46 47.16 48.3 22.49 50.29 51.12 35.52 26.53
Plantri 11-1135: |Aut|= 4 Vdeg = [0 5 2 4]
3.10 17.2 18.9 6.21 13.24 27.16 28.1 22.31 32.5 34.19 8.35 37.30 23.38 12.39 41.26 42.15 43.0 29.44 36.45 46.7 20.47 48.33 4.49 11.50 51.40 52.25 53.14
Plantri 11-1155: |Aut|= 2 Vdeg = [3 0 3 5]
11.0 6.13 19.4 2.21 26.7 12.27 1.28 29.22 16.31 33.18 34.3 20.35 5.36 37.14 24.39 41.10 8.43 44.25 38.45 15.46 47.32 48.17 30.49 23.50 51.40 52.9 42.53
Plantri 11-1210: |Aut|= 2 Vdeg = [0 3 6 2]
1.10 15.6 3.20 13.22 25.0 26.9 28.17 4.31 21.32 12.33 35.24 37.8 27.38 39.18 41.30 5.42 14.43 23.44 34.45 46.11 47.2 48.19 49.40 50.29 16.51 7.52 36.53
Plantri 11-1232: |Aut|= 2 Vdeg = [2 1 4 4]
14.3 5.16 18.11 23.2 15.24 4.25 26.13 29.20 31.0 6.33 17.34 35.12 27.36 38.21 39.30 8.41 43.10 19.44 28.45 37.46 47.22 48.1 49.32 7.50 51.42 52.9 40.53
Plantri 12-7571: |Aut|= 2 Vdeg = [1 4 1 6]
15.6 1.20 11.22 24.9 27.18 4.31 12.33 23.34 35.10 36.21 37.2 39.30 5.40 14.41 43.8 25.44 45.0 46.19 47.28 16.49 7.50 42.51 52.13 32.53 54.3 55.38 56.29 57.48 17.58 26.59
Plantri 12-7572: |Aut|= 2 Vdeg = [1 4 1 6]
11.2 5.14 17.8 19.0 7.24 16.25 30.9 31.18 21.34 13.36 4.37 39.28 26.41 42.15 43.6 44.23 33.46 20.47 1.48 10.49 50.29 51.40 27.52 38.53 54.3 55.12 56.35 57.22 45.58 32.59
Plantri 12-7593: |Aut|= 2 Vdeg = [0 3 6 3]
11.0 5.16 20.1 21.12 6.25 17.26 28.3 30.13 31.22 9.34 18.37 27.38 39.4 40.15 42.23 43.32 10.45 35.46 48.7 24.49 50.41 14.51 52.29 2.53 54.19 36.55 47.56 57.8 58.33 59.44
