Anselm Levskaya's polyHédronisme website makes it easy to test which of the classic Conway polyhedron-building operations conserve cursality. The 92 Johnson solids have varying amounts of symmetry, about 12% are unicursal: the knotted Johnson solids are J1, J2, J3, J5, J7, J8, J12, J13, J16, J20, J54. The links below are to the polyHédronisme viewer.
Johnson solid J54 makes a good test case: it shows that among the 16 polyhedron-building operations, k, a, g, d, r, e, b, o, m, t, j, s, p, c, w, q, only d and r (dual and reflect), as is well known, conserve cursality. (Actually, not just reflect, but any embedding of a graph in the sphere will have the same cursality as any other.)
It is known that, if a graph is 3-regular, t, the truncation operation, conserves cursality. Since k = dtd, and d conserves cursality, it follows that, for a graph that is triangle-faced, k (kis) conserves cursality. At best, if a graph is 3-regular or triangle-faced there are just three operations that conserve cursality: dual, reflect (or likewise any alternate embedding in the sphere) and t or k respectively.
Here is the cursality of each Johnson solid.
J1: 1, J2: 1, J3: 1, J4: 4, J5: 1, J6: 2, J7: 1, J8: 1, J9: 5, J10: 5, J11: 2, J12: 1, J13: 1, J14: 3, J15: 2, J16: 1, J17: 2, J18: 3, J19: 2, J20: 1, J21: 2, J22: 5, J23: 4, J24: 3, J25: 8, J26: 2, J27: 2, J28: 6, J29: 2, J30: 2, J31: 2, J32: 7, J33: 3, J34: 4, J35: 2, J36: 4, J37: 2, J38: 2, J39: 6, J40: 3, J41: 3, J42: 4, J43: 4, J44: 2, J45: 2, J46: 2, J47: 3, J48: 4, J49: 3, J50: 2, J51: 4, J52: 2, J53: 2, J54: 1, J55: 4, J56: 2, J57: 3, J58: 2, J59: 2, J60: 3, J61: 4, J62: 3, J63: 2, J64: 2, J65: 4, J66: 2, J67: 4, J68: 3, J69: 4, J70: 5, J71: 5, J72: 4, J73: 12, J74: 4, J75: 8, J76: 3, J77: 3, J78: 4, J79: 4, J80: 4, J81: 6, J82: 4, J83: 3, J84: 3, J85: 2, J86: 3, J87: 2, J88: 2, J89: 3, J90: 3, J91: 4, J92: 6,
No comments:
Post a Comment