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.