Joining link components to form knotted deltahedra

The tetrahedron is a Tait graph of 3-component link rather than of a knot.

Many interesting deltahedra, foremost among them the triangle-faced Platonic solids, are not knotted, so it is interesting to look at joining link components to yield knots. [Following Q. Xing, E. Akleman, J. Chen, and J. L. Gross in “Single-cycle plain-woven objects,” 2010 Shape Modeling International Conference, IEEE, pp. 90-99, 2010] we find that a link with n components can be converted to a knot with n-1 small edits.

For example, if one edge of a Tait graph carries two different link components (every edge in the tetrahedron above qualifies) simply subdividing that edge with a 2-valent vertex joins the two link components into one. The edited Tait graph will no longer be a triangulation, but this is easily fixed. We can add a pair of parallel edges between the two neighbors of the new vertex (because this is a graphical Reidemeister move that does not alter cursality), and then we change the embedding so that the two parallel edges straddle the subdivided edge (change of embedding does not alter cursality.) Now we have recovered a triangle-faced Tait graph with one less component, albeit no longer 3-connected nor simple, which we can nonetheless weave.

In the tetrahedron, performing this edit on any two edges that share a vertex joins the three link components into one. Road code: 3.6 8.1 9.12 2.13 14.7 15.0 17.10 11.18 19.16 21.4 5.22 23.20

The tetrahedron becomes knotted by the addition of two ears.