Table of Contents
We feel that knot theory has to do with our project.
You can have knots and links embedded on those graphs. Here are some preliminary screenshots (using our KnotViewer utililty) showing a trefoil knot on a 'dode_tor_6' torus and the projection, but in the latter unfortunately the three crossings are not identified at one glance:
Figure 10.1. KnotViewer Trefoil
KnotViewer displaying a trefoil on a graph of genus 1 and in projection
The trefoil knot can be embedded on a torus without any crossings left.
So called (p,q) torus knots and their generalizations are discussed here (see for example [Ada94] and [Hat02]). On a torus the numbers (p,q) are retrieved counting intersections with standard coordinate lines. The link crossing number is then min(p(q-1), q(p-1)).
Well, we think that this is actually using a word in the generators (a,b) to specify how the path should go (see the section called “Holonomy Groups”). Since we need another dimension to specify if a crossing is over or under we could focus on 2 levels (where collisions happen) and moreover on alternating knots. Of course other rules (over over under under, over over under over under under, over over under under over under …) and more levels are possible as well. On a torus (that is to say an Abelian group) you can encounter the problem as well, because the path will cut itself as soon as after one round there is no perfect matching for the overall movement; the discreteness of the steps (of the generating word) is decisive, with an infinitesimal setup no crossings would occur on a torus.
Here (Figure 10.2, “Dode Tor 6 knot”) is some screenshot with generator "aab".
Figure 10.2. Dode Tor 6 knot
"aab" knot with 6 crossings, writhe -2 and 3 levels on the "Dode Tor 6" graph
Exercise (worth 14 points): Prove, refute or declare as undecidable the following conjecture: all knots can be embedded as an alternating word(over a,b)-knot on a discrete 2-manifold with suitable 'vector' group and genus (torus knots with Abelian group are only the easiest example).
Maybe the conjecture should even postulate embeddings on graphs without any crossings left (like the trefoil on a torus), does such a graph always exist for any possible knot? Well, or we could settle the crossings question as follows: just take a starting point for your path, and as it happens in time, if you cross your old trail, then it's always now an over for u, until you arrive at the start again, then you must identify with the rope's start. But perhaps in the future we should pay attention to the torsion/cover thing and same as a single atomic rot gives a crossing and not a glue, a complete spin = rotpoly should not allow a glueing as well …
It is known (see for example [Man04]) that every knot (and link) is n-embeddable (on surface of genus n that is), every alternating knot is toroidal alternating, and every fast-alternating knot is toriodal alternating as well. So this goes into the conjecture's direction at least (it's not that much of a difference anyway, only discrete case now).