It is the geometry, not the topology of the fig. 8 knot that is utilized here. Any S shaped knot can serve as a base for such a loop. So, instead of knots topologically equivalent to the fig. 8 knot, we can use simpler S shaped knots, topologically equivalent to the overhand knot, or even to the unknot. All we need is,
a : to form, as a base, a geometrically S shaped knot on the standing part, and,
b : to pass the working end through this knot two or three times ( two times, as at the # 0 and #4 loops, three times, as at the # 1, #2 and #3 loops).
If we could possibly achieve essentially the same thing using a simpler base knot, which would be much easier to untie, why should we use the fig.8 knot ? An S shaped overhand knot would probably produce equally secure loops.
Without probing distinctions between "geometry" & "topology"
in knotting, here, suffice it to say that this set of knots arises
from working the commonly given fig.8
into an altered form and then <... --variations of the set>.
Yes, one can do such general working with other bases,
but so what? (Casting the Eskimo bowline
into this shape I think
distorts that knot's real geometry
more so than does this set's
use of the fig.8
. The "S" is hardly essential, either, generally
but it is particular to this set, and the idea set off upon in exploration.)
set has a characteristic enclosure of the SPart's
eyeleg exit/entry by the turn of the SPart that is lost, e.g.,
in that Eskimo bowline
I see (slowly) that a version that I verbally sketched in the OP's
thread is his #0 with the tail's path reversed; it's not all so clear
to me which should be preferred --#0 is surely better on ring-loading
And both of these I think are better bets at easier loosening than
others --in that their brief entanglement of the tail allows it to be
used qua lever for loosening--, though I've not played much with
the full set.