Weaving deltahedra without knotology

The 9-vertex z-knotted tetrahelix realized as a road code basket.

This is the same deltahedron, Plantri 9-12, whose dual was illustrated in an earlier post. The road code weaving strip was cut from this .svg file. As is evident in the .svg file, the strip is cut with a slight taper so there is no cummulative misalignment due to the thickness of the three layers. The strip may need to be pulled taut during the winding process to stay in registration with the layers below.

Each fold embodies a crossing, so each fold bears two numbers. The circular holes allow the numbers on the lower strands to remain visible in the finished basket; each elliptical hole is a slot for the tabs at the base of each triangle. The triple layering along with the reduced number of triangles (compared to knotology) result in a very stiff construction.

As in the knotology version, the weaving code can be recovered by collecting the ordered pairs of numbers exposed on the finished basket. After collecting all the ordered pairs, rank them by lowest high number, e.g., in this example (26, 15) comes before (21, 28).

Some tetrahelixes are knotted (unicursal)

Any tetrahelix on 3+6n vertices is unicursal, 9 vertices in this example.

Any tetrahelix on 5+6n vertices is unicursal, 11 vertices in this example.

The tetrahelix, or Boerdijk-Coxeter helix, is theoretically of infinite length but naturally truncates to a finite number of tetrahedron units. Some lengths of tetrahelix are knotted, some are not. This is easily explored in SageMath, since the tetrahelix is built up by a simple recursion that can start from a triangle:

def append_to_last_3v(my_graph):
	next_v = my_graph.order()
	#return (next_v)
	G.add_vertex(next_v)
	G.add_edge(next_v, next_v-1)
	G.add_edge(next_v, next_v-2)
	G.add_edge(next_v, next_v-3)
	#G.show3d()

#EXAMPLE/////////////////////

# Start from a triangle
adj_list = {
	0: [1, 2],
	1: [0, 2],
	2: [0, 1]
}

G = Graph(adj_list)
#Odd no. of spanning trees == unicursal
mu = G.spanning_trees_count()%2
print(G.order(), mu)

for i in range(8):
	append_to_last_3v(G)
	mu = G.spanning_trees_count()%2
	print(G.order(), mu)
	
#G.show3d()

Output:
	
3 1
4 0
5 1
6 0
7 0
8 0
9 1
10 0
11 1

Though one recursion generates the whole sequence, we may prefer to see the two subsequences as descendant from two different unicursal triangle-faced plane graphs: the triangle leads the 3+6n sequence and the dipyramid (a.k.a., bipyramid) leads the 5+6n sequence. The significance of being unicursal is that these lengths of tetrahelix can be realized in road code weaving.

"House of n gables," the operation dual to "t3...n4"

The result of "house of n gables" operating on Johnson solid J20.

Johnson solid J20

A view of dt3n(3,0.5,0.3)n(4,0.5,0.3)dJ20 from the other side.

A view of Johnson solid J20 from the other side.

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.

P5

dt3...n4dP5

J16

dt3...n4dJ16

A polyhedron operation that preserves unicursality for general face spectra

In polyHédronisme, this is t3 n(10,0.5,0.3) n(5,0.5,0.3) n(3,0.5,0.3) n(4,0.5,0.3) J20. It is unicursal just like its base polyhedron, Johnson solid J20.

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.

Demonstration that t3n...n4 is unicursal for any face size: l to r, triple every edge in the graph (inverse III); doubly subdivide the arcs internal to each face (inverse IV); connect these new vertices peripherally with digons (inverse III); delta-to-Y the triangles thus created (V).

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,