Those are pretty cool ideas! You're coming at it from a completely different perspective than I am, and solving different problems (which does not make your solutions in any way inferior). You're looking at "how can I cover this thingy here with a pretty THK?" and coming up with ways to make one that fits. The method in your first link basically boils down to "guess at some number of bights and leads, 'draw' a grid using your first pass, then follow the grid." This has distinct advantages over the more "scientific" method of deciding on a knot, drawing the grid, and wrapping that around your foam core or whatever. You have more flexibility and can intuit the shape of the final knot better; you're not limited to the strict 45? grid normal graph paper (or most graphing programs) would limit you to. I think Ashley also mentioned something about using a first pass (of different string) to form a "diagram" or map for a THK, but that was, again, after working out the numbers in advance. Your way of using it essentially to decide the knot you're tying is clever.

From what I can tell, your second link involves THKs with multiple strands (as well as various different weave patterns), which is also pretty neat, and you get some great patterns that way.

I was sort of taking a more mathematical approach, not a practical one (I'm not a really practical person, it seems.) I've probably tied literally thousands(?) of THKs by now... and untied 99% of them. I carry string around and tie THKs around my hands on the train and at work, some of them huge and elaborate... and then untie them and start again. So I'm not focussing on actually creating something for a purpose, it would seem. I was thinking of a "purist's" subset of THKs: single-stranded, over-one-under-one, same number of bights on both sides, etc. Or another way, in the form of a French braid (over-1-under-1) that goes around in a circle. And I was thinking, how can I tie these? My motivation was more theoretical and mathematical, I guess: trying to see what could be done to form these particular knots, of various dimensions, but not necessarily for a particular purpose.

Let me see if I can explain what I have, in extremely broad strokes:

Basically, there are certain classes of knots you tie more or less "directly": all one-bight knots, all two-bight knots, all two-lead knots (multiple overhand), and all three-lead knots (braiding around). Also two more classes that involve some "tricks", but can similarly be tied sorta-directly: Nx(kN+2) and Nx(kN-2) knots. I have methods for tying these.

Once you have a THK, there are ways to "build" it into a higher-order one. They're pretty well-known, at least in simple cases, but I don't think anyone else has worked them all out in the detail I have. Each knot can be built in two directions, plus there are sort of higher-order builds (not the same as building and then building again). Suffice to say, it turns out that with the building techniques and the classes of knots described in the last paragraph, you can get to *everything*. I have a little program that lets me make a pretty table of the knots, with each one showing the knot it builds from and how... Here, I'll upload a small example. The real one will be bigger and with better detail, but you can see in this one, each knot shows in color above it the knot it's built from. The ones labeled in purple are in the "base classes," others have a label naming a smaller THK, and the color indicates how to build. It works out, you'll see.