If there were no friciton, how would you ... knot?!
.
Elementary, my dear dL...

By using the pair of the eye legs ( of the single loop s main bight ) as a "pivot" through the two parts of the knot. This pivot does not need friction to remain in place, because it is loaded by both sides. In any properly called "single loop knot", we should keep in mind that there are four, not two limbs : the two standing ends and the two eye legs. Therefore, we have a two-parts knot. If we imagine all of the limbs loaded, we can see that the knot could remain knotted, indeed, even if there were no friction.
I had to use this simple "thought experiment" in order to clarify what I had meant by this "
non-functioning elements" neologism - because any segment simply penetrating and/or interwoven within the knot s nub does affect its form, by its mere spatial presence within the knot, by the bulk of its body ( so, it makes a difference, regarding the shape and place of the
functioning elements as well ). Those pairs of bights you had tied to each other with the reef or the ABoK#1452 ( in fact, with whatever other bend you can think of

), do alter the flow of the tensile forces within the functioning elements, although their presence is not necessary for the knot - i.e., if and while the knot is loaded by its four limbs, it will remain knotted, even in their absence, albeit in a very different form.
Let me try again, using a second mental picture, which could possibly clarify the first one I have used previously.
Imagine there is no inherent friction whatsoever in any segment ( "
functioning" or not ) of the rope on which a single TIB eyeknot is tied on. If you load ALL its four limbs ( the two standing ends and the two eye legs ), pulling them from whatever direction you wish, can you untie the knot ? No. What you will be left with will be two still entangled segments, which can not but remain entangled, not because of the geometry, but because of the topology. However you manipulate the ends of those segments, without altering the topology of the whole knot ( i.e., without re-tuckings of these ends through the knot ), you will not be able to disentangle them. This will tell you that one, at least, of the two segments is not TIB : what was TIB was the initial whole knot, but now you load/pull the ends of its four loaded limbs, one, at least, of its two parts can not become both topologically equivalent to the unknot, so you can not disentangle them from each other without altering the topology of the knot. Therefore, you do not need friction to be able to "knot" a four ends tangle of two ropes, simply because it is not but a non-TIB bend of those ropes.
I will attempt a definition of the "
functional" and "
non-functional" elements of a knot: If all its limbs are loaded by ( say, just to simplify the mental picture ) an equal load/force, and we have reached a mechanical equilibrium, the "
non-functional" elements would be the elements whose rope length would have been consumed, they would have been swallowed inside the still knotted tangle, where they would have been almost straightened out. ( This "almost" means that what is left of a long "
non-functional " bight, like the one of the two that are tied by the Reef or the ABoK#1452 of the suggested counter-example, can still remain as an open, widely curved rope segment, but not the convoluted within its "twin" double line of the initial
"non-functional" bight. )
the possibility of tying with bights qua ends is interesting--although in this case one would be just solving a problem, and otherwise have an awkwardly large knot.
You may mean that you can tie two "Gordian" knots, each with the one of two ( or four, six, etc.) symmetric bights and the one end of the two ends of the whole knot, which do not need friction... and so they are non- "functional", although they are not condemned to disappear in the absence of friction forces. Firstly, this knot would be awkwardly complex, as you say - there are no "simple" Gordian" knots that can be used as parts of a practical knot. However, I believe that such a knot can not even exist and be symmetric, because any connecting segments ( the "pivot" of the main bight, or other bights extending from those Gordian parts and encircling the pivot and/or each other, would "break": the symmetry : one of them would still have to be "above" or "below" the other, etc.