Why teafork works

The graphical Reidemeister moves (from Jiang, Jin and Deng, "Determining the component number of links corresponding to triangular and honeycomb lattices," 2012.)

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.)

The graphical Reidemeister moves (from Noble and Welsh, Knot Graphs, 2000.)

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.

From left to right, a demonstration that truncate, t, preserves cursality in a 3-regular plane graph: a vertex in the graph; same vertex after inverse-IV is applied to all edges in the graph; same neighborhood after V is applied.

Dually, we can demonstrate that kis, k, preserves cursality when applied to a 4-regular plane graph (figure below.)

From left to right, a demonstration that kis, k, preserves cursality in a face-3-regular plane graph: a face in the graph; same face after inverse-III is applied to all edges in the graph; same neighborhood after V is applied.

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.)

A demonstration that k4t preserves cursality in a 4-regular plane graph: a) a vertex in the graph; b) same vertex after inverse-IV is applied to all edges in the graph; c) same neighborhood after inverse-IV is applied to the central vertex; d) after V is applied to the two Y vertices; e) after inverse-IV is applied to the central vertex; f) after V is applied to the outer vertices in "Orion's belt."

Dually, we can demonstrate that cursality is preserved in a face-4-regular graph when teafork, t4k, is applied (figure below.)

A demonstration that t4k preserves cursality in a face-4-regular plane graph: a) a face in the graph; b) same face after inverse-III is applied to all edges in the graph; c) same neighborhood after inverse-III is used to bisect the face; d) after V is applied to two triangles; e) after inverse-III is used to bisect central quad; f) after V is applied to the resulting two triangles.

Teafork: a polyhedron operation that preserves cursality on {3, 4}-face regular polyhedra

The teafork operation (t4k) applied to Johnson solid J8

Expressed in polyHédronisme algebra, the polyhedron-building operation t4k (teafork) preserves cursality when applied to polyhedra that have only triangles and quads as faces. In effect, teafork augments each triangle face with a pyramid and each quad face with a frustum. Since knotology weaving builds pyramids on triangle faces and weaves quad faces flat, an idea of the resulting knotology basket is given by applying k3.

Polyhedron-building operations that conserve cursality

Snubbed Johnson solid J16, sJ16, is unicursal.

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.

J1

J2

J3

J5

J7

J8

J12

J13

J16

J20

J54

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,

Cursality of some classic deltahedra

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.

Tetrahedron 3

Octahedron 4

Icosahedron (1f geodesic dome) 6

J12 1

J13 1

J17 2

J51 4

J84 3

dtI (Buckyball's dual) 10

u2T (2f tetrahedron) 6

u2O (2f octahedron) 8

u2I (2f geodesic dome) 12

u3I (3f geodesic dome) 18

u2J12 (2f J12) 2

u3J12 (3f J12) 3

u2J13 (2f J13) 2

u3J13 (3f J13) 3

Viewing trirectangular stellations of deltahedra

The trirectangular stellation of Johnson solid J51.

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.

Johnson solid J51; Trirectangular stellation of J51.

Johnson solid J84; Trirectangular stellation of J84.

Realizing (and stellating) deltahedra

Not every triangulation of the sphere can be realized as a deltahedron, that is, a solid having faces that are all equilateral triangles. Furthermore, if such a deltahedron does exist, its trirectangular stellation (which is what knotology weaving produces) may have faces that intersect. The former problem has been investigated in Tsuruta, Naoya, et al. "Enumeration of deltahedral graphs with up to 10 vertices." Proceeding of the 16th International Conference on Geometry and Graphics (ICGG2014). The images shown here have been marked up from that paper.

Basically, from our road code weaving perspective, three things can go wrong with a triangulation on the sphere: it might be multicursal (marked up with an X in the figure above), it might be non-deltahedral (an N in Tsuruta et al.'s code for the candidate deltahedron), or it might be non-stellate-able using trirectangular pyramids (marked up with a strike-through).

The first condition can be easily tested in the abstract graph: the graph is unicursal (in all of its embeddings in the plane) if its spanning tree count is odd.

The second condition is difficult to prove, but Tsuruta et al. have calculated computer models of presumed deltahedra on a small number of vertices that are equilateral to within 1 part in 100,000.

The third condition is sometimes easy if there is a dihedral angle in the deltahedron that is not obtuse enough to accomodate trirectangular pyramids on each of the adjoining faces. A trirectangular pyramid has dihedral angles of 54.74 degrees at its base, so a dihedral angle in the deltahedron less obtuse than 2 x 54.74 = 109.48 degrees cannot accomodate stellation. Such an angle is noticeably more obtuse than 90 degrees, so any dihedral angle of 90-degrees or less is disqualifying on sight.

It is worth noting that Tsuruta et al. use the same Plantri-based naming convention as here, but they count indexes from one rather than zero. Thus the pentagonal dipyramid is 7₄ in their nomenclature, but 7-3 here.

Also Tsuruta et al. exclude deltahedra candidates having dihedral angles of exactly 180 degrees; from a basketry perspective there is not really a need to object to these passages of flat weaving. For example in the figure below, Tsuruta et al. exclude 5 of the 14 candidate deltahedra on 8 vertices (N in their code stands for non-deltahedral) because of they 180-degree dihedral angles. Coincidentally we would exclude all the same (and more) for not being unicursal (marked up with X).

A knotted kagome "nanotube"

Among the smallest knotted (unicursal or single-cycle) kagome baskets are some elongated small-diameter shapes reminiscent of carbon nanotubes. Shown above is the medial graph of Plantri 15-253174, which has the following road code: 10.17 4.21 24.1 27.6 31.12 14.33 8.37 2.41 42.23 43.0 25.44 46.39 48.19 29.50 52.35 54.15 32.55 13.56 57.34 53.58 59.16 11.60 30.61 51.62 63.36 9.64 65.18 49.66 28.67 7.68 69.38 47.70 71.20 5.72 26.73 45.74 75.40 3.76 77.22

This shape can be lengthened 6 vertices at a time. Its shorter predecessor is Plantri 9-12 of the previous post. The endcaps in both baskets are the same, the middle section, which in the 3-regular Tait graph is all hexagonal faces, gets longer 6 hexagons at a time. In carbon chemistry this would be a non-classical fullerene since each end cap has 1 triangular, 1 square, and 1 pentagon face. The face histogram for the 3-regular graph is [0 0 0 2 2 2 9].