A KnotTyer (1) diagram of the Gordian bend, presented at Re#12 :
http://igkt.net/sm/index.php?topic=3610.msg20970#msg20970 Here is an attempt to use this bend as a counter-example against the (false) claim that : "
Topology determines Geometry" :
Let us suppose we do not have access to any of the four ends of a bend. We can loosen this rope tangle, re-arrange it, and then tighten it again. Will we get the same knot ? Nooo! We may well get another knot, topologically equivalent but geometrically quite different. So, topologically identicall tangles, when they are dressed differently, generate geometrically different knots. Moreover, there can be cases where the initial tangle, when tightened, can lead to a knot, or not. The Gordian bend shown here has the same topology with two not-linked open loops. We can loosen it, without changing the topology of its links, and tighten it again. Depending upon the particular dressing, at the end of the tightening we may get a secure tight bend, or two loose unknotted loops ! The interest thing is that this fact is not depending upon the physical characteristics the ropes. Any bights with inaccessible ends, however slippery the ropes they are made of would be, can be linked by this bend.
in other words;
" At the attached files, you can see a 2D diagram and pictures of a bend I call " Gordian" . It is a link between two closed bights/loops, where we do not have access to their one tip. ( I have drawn these inaccessible "ends" as been warped around two poles, to show that we cannot use them, but also for decorative purposes...)
The most interesting/important thing is that this bend does not use friction to work ! It will work even if the ropes well 'ideal" - in the same sense we call the abstract mathematical structures that some mathematicians and physicists study, as " ideal knots" . The two links will not slip the one through other, because in their way to do this, they encounter obstacles imposed by the volume, the bulk of knots tied on the ropes. In this particular bend, there is a double line overhand knot tied on the one link (blue/white), and the accessible bight of this link is warped around it and it cannot overcome it. The bight of the other link (red) is trapped in this tangle.
Of course, we can untangle the bend, in the same way we have tied it in the first place. However, once we have tightened the bend, to be able to untie it this way, we have first to set the bend a little bit loose, to un-tighten it for a while. Then we can pass the bight over the bulk of the double line overhand knot, pass it through out the other s link bight, and release the tangle of the two closed loops.
So, once the bend is started to be tightened, and this tightening does not stop and is not reversed for a while, the bend can not be untied in an way - unless you cut the rope ! ( This is the reason I have called it " Gordian bend' ) Also, do not forget, this is independent of the low or high friction of the ropes, it will have happened even if our ropes were infinitely slippery, i.e. if they were "ideal' ropes.
The 2D diagram of this bend is the one shown. at the attached file. However, the topologically equivalent 2D diagram, is just two not-linked closed loops ! You can pass from the one diagram to the other, without having access to the ends of the ropes.
So, topology DOES NOT always determine the geometry of the knots, even if these knots are "ideal" knots."
1)
http://daveroot.co.cc/KnotMaker/