Stay in your lane: 1 in 3 spherical triangulations are z-knotted

In a triangulation of the sphere, a triangle strip can be seen as sort of topological geodesic, an attempt to travel as straight as possible within the given constraints. Notice that both the sequence of shared triangle edges (black) and the edges of the dual graph that connect the triangle centers (green) are zigzag paths, alternating hard-right with hard-left turns.

In a triangulation of the sphere, a triangle strip (above) is a sort of topological geodesic, an attempt to traverse the triangulation along a path as straight as possible without regard to lengths and angles.

Suppose we start at an arbitrary triangle in the triangulation and head off in one of the three possible directions, and from then on we stay in our triangle lane. What happens?

Clearly the strip must eventually loop back on itself since the sphere has no boundary that the strip could end at, and the number of triangles is not infinite. What is the chance our randomly chosen lane will visit every triangle exactly three times and then repeat? Seems a long shot...but apparently not. The chance of this happening is about one in three no matter how complicated the triangulation. A triangulation with such a triangle strip is termed z-knotted (z for zigzag.)

About 1/3 of spherical triangulations are z-knotted (a.k.a., unicursal). The integer sequence of spherical triangulations on n vertices is OEIS A000109.

Most of the numbers in the chart above were first published in 2004 in "Zigzags, Railroads, and Knots in Fullerenes," by M. Deza, M. Dutour, and P. W. Fowler. Below is their chart for triangulations of the sphere.

Z-knottedness in trivalent polyhedra and their dual triangulations. Chart from Deza, Dutour and Fowler.

Our 'n' counts vertices in the triangulation, while the 'n' in Deza, Dutour, and Fowler counts vertices in the dual 3-valent graph: a triangulation with n vertices has a dual with 2n-4 vertices (see the OEIS link above,) so the two charts are identical as far as they go.

Deza, Dutour and Fowler also looked at the same question restricted to fullerenes which is equivalent to restricting spherical triangulations to vertex degrees 5 and 6. The triangulations dual to the largest fullerenes they investigated have (74+4)/2 = 39 vertices, and all vertex degrees in {5, 6}.

Z-knottedness in fullerenes and their dual triangulations. Chart from Deza, Dutour and Fowler.

In this restricted class of spherical triangulations they found a z-knotted fraction of 1970/14246 = 0.138, about 14%. So z-knotted triangulation on the sphere are quite common even if technological constraints on vertex degrees make them somewhat less common.

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).

Plantri 11-382, the 11-vertex z-knotted tetrahelix, realized as a road code basket.

Plantri 11-719, an 11-vertex z-knotted deltahedron with the same vertex degree histogram as Plantri 11-382.

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.