We still have not yet implemented the construction of geodesic manifolds in the computer. The local constructions work. What happens however is that when building up the manifold, we will have to rename the vertices as we locally do identifications which globally do not hold. In the case k=1, when we computed geodesics, we had no difficulty as we just could work in the frame bundle, the set of oriented maximal simplices of the host manifold G. We as usual work at first in the case k=2, q=3, where we construct geodesic 2-manifolds in a 3-manifold. The spider centered at x has three immediate neighbors u,v,w but these three neighbors already have an identification as there are two geodesics which pass through each of the 3 neighboring simplices we can go to. The geodesics will split up and produce the 6 legs of the spider.
This week, I thought to simplify the construction by starting at a point x and then produce all the 6 geodesics through x and globally draw them out. This produces the “clover”. Every geodesic of course is finite and so closed as we are in a finite setup. There is no other choice than a clover structure. We then identify the immediate neighboring simplices. This is no problem, as pointed out in the short presentation today, we had that already last year as we produce the global structure of all the geodesics in a manifold. Now, we have the task to pick an other point z on the clover (not immediate neighbors as we have ambibuity there about the orientation) and produce the clover construction at this point. In the example below, I did that with a point z in distance 2 away from x. After gluing the spider legs at x and z we have to identify also two points near u, the point between x and z. We have now a larger clover. Note that these loops are in a large manifold in general still very large loops, in general not homotopically trivial. We have to build more clovers.
In the example below I took as the host manifold the smallest 3 manifold there is, the 3-sphere K(2,2,2,2), which is the join of two C(4)=K(2,2) circular graphs. The loops one can see are not the minimal loops yet. It would produce negative curvature patches. I will have to continue programming this at a later point.
