Evidently, we have 4 more cases of bi-stable knots here (1). The A, B, BB, C and D crossing knot-based eye-knots can be transformed into the 1, 2, 22, 3 and 4 locked bowlines, and vice versa. Identical, topologically, knots can be dressed in totally different geometries, therefore, when they will be loaded, they will be loaded differently, and that means that they have totally different structures - i.e., they are totally different knots. Let us call any of those 4 eye-knots which is based on the crossing knot nipping structure "
unfolded", and the corresponding eye-knot which can be derived from it, without changing its topology, by re-arranging the strands of the ropes, "
folded" - meaning that when the collar around the eye legs of the former is "folded" over the knot snub, when it is pushed backwards, towards the Standing end, and becomes a proper "common" bowline s collar, the "unfolded" eye-knot is transformed into a "folded" one. Of course, as we can go from the crossing knot based eye-knots to the corresponding locked bowlines, we can also do the opposite, but since I had first tied the A,BB,C and D series and only much later the 1, 2, 22, 3 and 4 series, I prefer to think of them in that order, from the A-D to the 1-4. Also, from all those eye-knots. I prefer the A ( the "pet loop" ), which is a very stable, secure, good knot and also a very easily tied one as TIB, so it comes natural to me to think that the A generates the 1, although the 1 can also generate the A by the reverse transformation.
I want to notice that, in this thread, I have done nothing than the re-dressing of already existing TIB crossing knot-based eye-knots, which can work at least equally well as the locked bowlines derived from them by this "folding" transformation. It may well be the case that we would arrive at a stable, secure knot even if we would had started from an unstable, not-secure one. So, I may have missed some stable, secure locked bowlines, because their corresponding crossing knot-based eye-knots had happened to be somehow unstable and insecure, and so they had not caught my eye. The interested reader is called to try his own hand.
The moral of the story is this : When you see a collar around both eye legs of an eye-knot, try to "fold" it over the knot, by pushing it to the "back", to see if you can transform it into a proper collar around the Standing end - or, when you see a proper collar, try to "unfold" it by pushing it to the "front", towards and around the eye legs. You may discover yet another bi-stable knot pair.
1.
http://igkt.net/sm/index.php?topic=4201
