Well, for those knots, all SPATIAL regions that come into existence as the knot is made have no associating syntactic representation; but, using the T specification, all crossings that occur BELOW the S region are officially in the T region. See? Most of the knot (except possibly for the crossing that touches the standing region) can be considered to be in the T region, since it was formed strictly by the tail’s moving over and under itself. With the T specification, your notation will be able to accommodate the overs and unders of a tail’s crossing over and under itself, for as long a sequence as is necessary to finish a knot. Since a practical knot can be only so big, the “sequence” will automatically be limited to a practical length.
Considering the possibilities of sequence, selection, iteration, and recursion, I’d say that sequence is the best choice!
". . . is it more natural for most people to tie the overhand knot in such a way that the tail emerges above the L bight (and the standing end below the H bight)?"
Yes, the notation needs to be able to handle all of the possibilities equally well.
When making a Bowstring Loop, I go under and over and under; when making a Butterfly End Loop, I go over and under and over. As if that weren’t complicated enough, my father was left-handed, I am right-handed, and I use both hands when playing piano!
Consider adding to the notation a numerical syntax by which the SPATIAL regions that come into existence as a knot is made are automatically created (i.e., enumerated). Then, you’ll have some serious specifying power for any situation. It could be complicated, because you’ll need to formulate an algorithm that defines the creation of a type of “grid” system–the labeling of which will have to change dynamically–as a knot develops, since there could be a piling up of regions both vertically and horizontally as a knot develops. . . .
Something like . . . “row 1, column 1,” etc., in the form “(1,1),” etc. The meanings of these would change dynamically as the notational sequence grows. . . .
By the way, sorry for my taking this thread into so many other directions.
By the way, if you’re ever tying down a load for my ex, then I recommend you do her the honors of tying an overly complicated knot that requires you to pull out a knot book and study for awhile.
Although these final loop knots are very simple, they result in some of the most complex-looking knots I’ve ever seen; but, they work.
I find it very interesting that when the Overhand is removed, much of what I sensed as beauty is also removed: Somehow, an Overhand gives a type of body or skeleton to a structure, which, when removed, results in what can be seen as a body without a skeleton.
But it could just be that we are all conditioned to accept the familiar and to shudder at the unfamiliar.
