The polyhedron operation described in the last post, "t3...n4," has a dual: "dt3...n4d." The dual operation preserves cursality since d, does. The results can be roughly described as building a roof with n gables at each n-valent vertex. The links below to polyHédronisme give more examples on unicursal base polyhedra.
The previously described polyhedron operations are limited to certain constraints on vertex valence or face valence. The operation described here can deal with any face size. The idea is to augment every face with its frustum and then truncate the new vertices. This can be accomplished with polyHédronisme's set of operators though the expression must be tailored to the range of face valencies in the base polyhedron. Since the frustum augmentation step (it is called inset, n, in polyHédronisme) creates new 4-valent faces, the frustum operation on 4-valent faces must be done first. The frustum operations on {3, 5, 6, 7...etc.}-valent faces can then be done in any order. After all the faces have been inset, all of the original vertices are now even-valent, so none are valence 3, and polyHédronisme operation t3 only truncates the new vertices.
If 'n...n' can stand for any number of valence-specific n's, this operation is t3n...n4.
A Tait graph of a knot projection can be transformed to a Tait graph of any other knot projection by a sequence of unsigned graphical Reidemeister moves (see Noble and Welsh, below, and Jiang, Jin and Deng above.)
For example, using the graphical Reidemeister moves, we can demonstrate that, in a 3-regular plane graph, vertex truncation, t, does indeed preserve cursality (figure below.) We will use the Roman numerals of Noble and Welsh to designate the moves.
Dually, we can demonstrate that kis, k, preserves cursality when applied to a 4-regular plane graph (figure below.)
The final figures seem to leave some stray vertices and parallel edges laying about, but these are destined to be "used up" as the same operations are applied to neighboring vertices or faces.
Similarly, we can demonstrate that cursality is preserved in a 4-regular plane graph when k4t is applied (figure below.)
Dually, we can demonstrate that cursality is preserved in a face-4-regular graph when teafork, t4k, is applied (figure below.)
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,
Most deltahedra are not knotted (unicursal) and that includes most of the deltahedra linked to here—Johnson solids J12 and J13 are knotted. If cursality is small, doubling a few strategically chosen edges in the deltahedron can splice together the link components into a knot that can be road-code woven into nearly the same shape. In particular, if a deltahedron is the Tait graph of an n-component link, just (n-1) edges of the Tait graph need to be doubled to convert the link to a knot. Here are the component counts for the links described by the following deltahedra.
Icosahedron (1f geodesic dome) 6
J12 1
J13 1
J17 2
J51 4
J84 3
I only recently became aware of Anselm Levskaya's elegant polyHédronisme website. PolyHédronisme gives you an interactive 3D model of just about any polyhedron you can generate from a classical 'seed' polyhedron using Conway's polyhedron operators. By twiddling with the depth parameter of the kis operation, a deltahedron can be viewed in its (approximately) trirectangular stellation—which is how it would be realized in knotology weaving.
The eight convex deltahedra can be generated this way: tetrahedron (T), octahedron (O), icosahedron (I), and Johnson solids J12, J13, J17, J51, J84. Unfortunately none of these classic deltahedra is knotted (a.k.a., unicursal.) Any model you make on polyHédronisme can be referenced by URL as the links below demonstrate:
Tetrahedron; Trirectangular stellation of the tetrahedron.
Octahedron; Trirectangular stellation of the octahedron.
Icosahedron; Trirectangular stellation of the icosahedron.
Johnson solid J12; Trirectangular stellation of J12.
Johnson solid J13; Trirectangular stellation of J13.
Johnson solid J17; Trirectangular stellation of J17.
