Let me try to tell some / the same things with other words - just in case somebody out there is listening...

1. A "real", "physical", "practical" knot is still a knot, does not cease to be a knot. It remains a knot, so it has all the essential characteristics of a knot - otherwise it would not be a knot any more...

2. The #1, the number one, the most essential characteristic of a knot, is its topology. It is an "invariant" mathematical quantity, a quantity that does not change. It is inherent in the definition of a knot, of any and every knot. A knot that has not topology, is not a knot. A knot that has a different topology, is a different knot. A knot can change its topology only if it is cut, i.e., only if it ceases to be a knot.

From 1. and 2. , any and every intelligent being in this universe, be it an extra-terrestrial or not

, concludes that a "practical" knot has topology, and that this topology can not change - unless the rope of the knot is cut !

A "real", "physical", "practical" knot cannot be unknotted / "untied" due to one, at least, of the following two conditions :

1. Topology

2. Friction

A. Are there any "real", "physical" knots that can remain knotted/tied without relying on topology or friction ? Yes, there are, and they are called "Gordian knots". The first known example was offered by Pieranski et all (1), and I had offered two examples of "Gordian links"

( knots where the two parts are topologically equivalent to two connected but topologically unlinked closed loops ) (2).

B. Are there any "practical" knots that can remain knotted/tied without relying on topology or friction ? We do not know yet any Gordian knot that can serve as a practical knot, so we can suppose that, probably, there are not any. I had tied two "Gordian bends" (3), where two closed loops / slings are connected to each other without the need of topology ( they are not topologically linked ) or friction, but they are bulky and quite complex, so they can not be considered as "practical".

C. If a knot is already topologically linked to another knot, will it need friction to remain linked ? No, of course not, because topology is more than enough ! Topology can not change, so two topologically linked knots will remain linked, even if tied on the most slippery material of this universe ... For example, two linked simple closed loops, or two linked complex closed loops, will remain linked, however rearranged their rope segments might become, and however slippery the material on which they are tied might be.

D. A knot of which we hold both ends, for example an overhand knot, is topologically equivalent to a knotted closed loop, because its ends are considered inaccessible. By the same token, a hitch which we hold by both ends, is topologically equivalent to a closed loop, too. In the particular case of a hitch, which is usually tied around a bollard, pole, hook or ring, we say that the hitch can not be unknotted/untied/released, if it can not be unknotted, untied or released without going over / slipping out of the one or both ends of the object around which it is tied. So, in essence, we treat the bollard, pole, hook or ring as a closed loop, because their ends are considered inaccessible regarding the ability of the hitch to be unknotted/untied/released. Of course, we can use the accessibility of one or both ends to tie the hitch ( as it happens in the case of the Pile hitch, for example), but we suppose that the hitch itself can not be untied by going over / slipping out of the any end.

E. Consider a bend of which we hold the four ends, or a hitch whitcg we hold by the two ends ( in the case of a hitch, the other "link" is the object, be it a bollard, pole, hook or ring ). Those bends and hitches are topologically equivalent to connected closed loops, to slings. Now, those closed loops, those slings, can be topologically linked or not - i.e., they can be connected using their topology ( like two linked simple closed loops ), or friction. The difference is obvious : If they are topologically connected, they are condemned to remain connected, because, if their rope will not be cut, they will remain knots, and knots retain their topology, and the topologically knotted ( linked) knots will remain knotted, and should be considered as one rope-made object - friction has nothing to do with this fact. If they are connected due to friction, but they are topologically unlinked, i.e. equivalent to two unlinked closed loops/slings, they can be separated, and should be considered as two individual objects - that happens to be attached to each other due to friction.

D. I could not have imagined that the simple fact that a bend of which we hold by/ load the four ends ( a four-ends bend" ), or a hitch which we hold by its two ends, would NOT be obvious that they correspond to connected closed loops... So I had not understood the purpose of stating the obvious, that two topologically linked closed loops will remain topologically linked, without " relying on friction". If their ends are not accessible, two interlinked Cow hitches will remain interlinked, so the Cow hitch bend (4) does not " rely on friction", and a Cow hitch ( a Girth hitch ) does not " rely on friction". This has nothing to do with the presence or absence of friction, just as it has nothing to do with the presense or absense of anything else, this is a direct consequence of the fact that those hitches are knots, and knots retain their topology, and topologically linked knots will remain topologically linked, not because they do not rely/need friction, but just because they are knots, and knots can not change their topology - at least in a 3 dimensional universe !

E. The statement " a Girth hitch does not require friction", is equivalent to the statement " two topologically linked closed loops do not require friction to remain linked", is equivalent to the statement "two topologically linked closed loops do not require words or pictures or unicorns to remain linked ", is equivalent to the statement "knots are knots, and they can not cease to be knots, so they will never cease to be knots", i.e., it is a

**tautology**.

1.

http://www.maa.org/devlin/devlin_9_01.html2.

2.

http://igkt.net/sm/index.php?topic=3610.033.

http://igkt.net/sm/index.php?topic=3610.msg20970#msg209704.

http://igkt.net/sm/index.php?topic=1919.msg16429#msg16429