Bistable knots.

[X1]

  The topology of a knot does not uniquely determine its geometry. Topology does determine zero to several allowable *knotted* geometrical conformations - in other words it determines a *set* of materially *possible* geometrical configurations -so it does not lead always to one and one only knot.
  So, one and one only tying diagram can lead to two topologically identical, but geometrically quite different knots. In this thread one can see two such knots, a Pretzel-to-Pretzel bend ( where each of the two interweaved links of the bend is a Pretzel-shaped overhand knot, shown at the attached pictures of this post ) and the Hunter s X bend ( the well known Hunter s bend, where the tails are crossed before they leave the knot s nub, shown at the attached pictures of the next post).
   Are those knots two different knots ? Yes, I believe they are. Why ? First, because they look so different... as everybody can easily see by just looking at those pictures. Moreover, and this is the most important thing, they are different because in each of those two knots the standing parts follow very different paths in space, and so they are loaded very differently. If we had been able to "see" the flow of the tensile forces as they "run" along their carriers, the standing parts ( going from the 100% of the load at the standing end to the 0% of the load at the tail), there would have been no doubt whatsoever about this.
   Why does the one and the one only tying diagram lead to two different knots ? It seems that, starting from an initial configuration described by a particular tying diagram of a knot, and proceeding by pulling the standing ends, we can reach one and one only stable form of this knot- and not any other. As the knot shrinks, and its volume is reduced, it will arrive at some level of stability, and it will settle to one stable compact form, from which any further pull of its standing ends will not be able to change its geometry. To transform this one stable form of the knot to another one, one has to intervene manually, and pull or rotate the collars of the knot, that is, disturb the achieved balance. Then, starting from a different initial configuration, the knot will be able to reach another lever of stability, and it will settle to a second, different stable form.
   "Pull or rotate the collars of the knot". I have done something like this to the Pretzel-to-Pretzel bend, shown in this post, which is a very stable, compact and secure knot, and arrived at the Hunter s X bend, shown at the next post. So, those two knots are two stable different forms of one parent "bistable" knot - or two different stable knots.   

[X1]

   Here is how different the "same" knot can be : ( I have retained the legths of the tails as long as they were when the knot was at its "Pretzel-to-Pretzel" first stable form ).
   The Hunter s X bend consumes less material, and it has a smaller volume. However, this does not mean that the Pretzel-to-Pretzel bend is not compact - it is compactified as much as it can be, by the pulling of its ends. To be transformed to a Hunter s X bend, the knot should start shrinking from a different initial configuration. Then it can bypass the first level of stability, and settle to the second one. Of course, nobody can say anything about the relative security and/or strength of the two knots...

[Dan_Lehman]

Part of what this discussion shows is our stumbling about without
a good knotting science with articulated definitions and nomenclature
that captures that.  "knot", "topology", "tangle", "geometry", "structure",
"form", "end", "tail", "SPart", "loop"/"loop"/"loop...", and so on.

I once wanted the purely interlacedness --however shaped--
that is captured in mathematics' "topology" (though there
there is some question of equality among ... <what?>,
for which the "invariant" is the key (to be found!))
to be my "tangle"
from which one could then specify distinct "knots"
by means of loading/angles.
But I saw that my conception of "tangle" had already
assumed some sort of "geometry" of things from which
the particular "knots" followed easily.  I.p., the geometry
in which the tail of a bowline is stuck through a bight of
a slip-knot (overhand noose, actually, per loading) isn't
what I had in mind for the general net-knot/becket hitch/
bowline/sheet bend/Lapp bend/Lapp hitch/eskimo bowline
set of *knots* that are specifiable from the general tangle
given by a sheet-bend geometry.  --the mere specification
of loading (or angles for ends) wouldn't generat the desired
*knots* ; there would need to be some pre-shaping of the
*tangle* for the spec.s to work.

Now, math --at least some main body of theory-- works with
"closed curves" (if I'm recalling this correctly), to which one
doesn't find "ends" --points of applying load or orienting at angles.

 . . .

 - - - - - - - -

Re "Hunter's X bend" (maybe Edward never thought of this,
but Lehman & Asher did --Asher dismissing it, which IMO is
a great mistake!), I find it the preferable orientation of rope
to the popular version.  One can see in X1's 3rd image of it
how a part of a tail occupies space within a collar, and this
helps keep the knot from jamming; there is also a nicer,
more gradual curvature in the SPart, which suggests more
strength / less *hurt* to the rope under load.

Beyond the dramatic --and, I agree, *knot*-defining--
differences in *geometry* presented by the OP, are
ones of lesser degree, such as loading a fig.8 end-2-end knot
"strong form" with vs. without firmly setting it by pulling
on the tails : absent such loading, the SParts compress
into the nub severely; with proper setting (IMO), they
bear much more into their trace-twin tails's turns.
(And, even here, YMMV per materials/loading ... .)


--dl*
====

[X1]

"topology" (though there is some question of equality among ... <what?>, for which the "invariant" is the key (to be found!)

  I believe that "topology", in mathematics, is well-defined, so there will be no ambiguity of the equality/equivalence (or not) between the topology of any two "knots". If we do not wish to search for a "what-thing" that remains unchanged / is invariant, we can define two topologically equal/equivalent knots, as two knots where one can pass from the 2D diagram of the one to the 2D diagram of other by a sequence of 3 simple transformations - the "Reidemeister moves" (1)(see the attached pictures). Easier said than done !  In practice, it may be really difficult to see the equality/equivalence (or not) between two knots - as the recent thread on the TIB double loops has shown...(2)

my conception of "tangle" had already assumed some sort of "geometry" of things from which the particular "knots" followed easily. 

Beyond the dramatic... differences in *geometry* presented by the OP, are ones of lesser degree

   Geometry should always come after the topology. Moreover, geometry may be similar, but it can never be identical (as topology can be). The mere change of the magnitude or the angle of the loading of one or more end(s) will change the geometry. However, in this thread I am not talking about the "lesser" differences such changes impose... I am rather talking about similar geometries, clustered around two stable "dramatically" different configurations. (I am not aware of any tri-stable knots.)

--the mere specification of loading (or angles for ends) wouldn't generate the desired *knots*
   there would need to be some pre-shaping of the *tangle* for the spec.s to work.

  Right ! Neither would the mere specification of the topology... That is what I have called as "different initial configurations". Two different initial configurations, although topologically equal/equivalent, can lead two distinct knots. The two knots shown in this thread offer an almost perfect example of this situation.

1.  http://en.wikipedia.org/wiki/Reidemeister_move
2.  http://igkt.net/sm/index.php?topic=4168.0

[X1]

  The uR-lL bend presented at (1)(See the attached pictures) can also be considered as a bistable knot - although the differences between the two stable forms are much less pronounced.

1.  http://igkt.net/sm/index.php?topic=3086.msg18722#msg18722

[struktor]

The (7,2) torus knot has been tied on a long, pefectly floppy rope.
SONO, an algorithm developed in the Poznan University of Technology,
simulates a process, in which the rope is continuously shrinking
what forces the knot to change its conformation.
Notice the symmetry breaking that occurs during the process.
http://www.youtube.com/watch?v=rzdRrs8OFjU

http://etacar.put.poznan.pl/piotr.pieranski/SKRETKI.jpg
http://etacar.put.poznan.pl/piotr.pieranski/IdealTwistedPair.html

http://etacar.put.poznan.pl/piotr.pieranski/New%20Scientist%20Round%20the%20twist.htm

Struktor

[X1]

  the rope is continuously shrinking
  forces the knot to change its conformation.

  You mean that, eventually, even these bistable knots will be forced to settle in one and one only stable final form, where the rope length will be minimized - and that this process will be a continuous one.
  Those simulations do not take into account the friction forces. In the "ideal" friction-free world, it is true that any knot will shrink to one and one only form, if it is tied on an "ideal", friction-free "material". ( The only exceptions are the rear and very complex "Gordian" knots, studied by Pieransky ). However, in the "real" world, topology does not define geometry in a unique way. The Pretzel-to-Pretzel knot will not shrink and become the Hunter s X knot, however forcefully one will pull its ends. It seems that friction places another energy "barrier", which separates the continuous process of the shrinking of this Pretzel-to-Pretzel knot, from the shrinking of the Hunter X knot. ( For the uR-lL bend, I am not so sure...) 
 

[struktor]

  You mean that, eventually, even these bistable knots will be forced to settle in one and one only stable final form, where the rope length will be minimized - and that this process will be a continuous one.

http://arxiv.org/pdf/1104.0489.pdf

Three minimum exist:

1. Average rope length and the slope of an ideal knot.
2. Best volume packing knot.
3. Energy minimizing knot.

[Luca]

Hi all,

A bend topologically equivalent to the "standard"(-x)Hunter's bend(obtainable without any kind of inversion of the roles of the standing ends and the tail ends)
                                                                                                                        Bye!

[xarax]

   Congratulations, Luca. Nice knot !
   So, the Hunter s bend, where the tails are "parallel" = not-crossed, can be transformed into this bend, where the tails are crossed. It looks like a two-strand / two-fold Mathew Walker s knot ( ABoK#776)-( M. B4) with crossed tails - and I believe that, due to that crossing, it is more stable than the round / compact, very symmetric, but somewhat unstable and not easily dressed Mathew Walker s bend. However, in such simple knots, even a minor transformation, like the crossing or uncrossing of the tails before they exit the knot s nub, can have a huge impact - so neither the topological identity nor the geometrical similarity plays any important role : all those bends are almost as different as they could had been ! If nobody has seen this knot published somewhere, your post should be transferred to the "New Knots" section.
  See some pictures of this bend, taken from different angles. I should also notice that your fourth picture, although it is meant to reveal the structure of the knot by showing it in an "exploded" / loose form, it is not very helpful in this - and it may even be misleading, if one uses it to actually dress the knot. Personally ( and especially with very simple knots, as this one ) I prefer to show them only in their dressed, compact/tightened form, and then ask from the reader to "decipher", and then tie and dress them by whichever method he finds more suitable - than to show a loose knot, which is something that is not corresponding either to the topological diagram with the minimum number of crossings, or to the final geometrical form of the knot - and it can not even be used as a tying/dressing diagram. ( It is very difficult to stabilize all the interweaved segments of an evenly / geometrically correct "exploded" real knot, and then take pictures of it ! - so I prefer to avoid the attempt...)

[Luca]

Thank you xarax,and also for the photos!

I had not thought of a resemblance to #776, in effect a relationship should be (and in fact the Hunter's bend,with the tails crossed the other way with respect to the Hunter's bend that you illustrate in your second post,is topologically equivalent to #776,and #1408/9);I personally noticed a sort of Carrick-way in the shape of the bend.
Regarding the fourth pic in my post, I was actually going to try to do another more correct one, but then I noticed that,photographically speaking, is the thing that comes closest to the decency of all the photos I've posted so far...
Finally, returning to the topic of this thread, I would give a look to #1408/9 and the uR- lL bends at reply #4...

                                                                                                                                   Bye!

                                                                                                                       

[Luca]

Hi,

A pair of steps which illustrate the transformation from ABoK #1409 to the uR-lL bend.

[Luca]

The Ashley bend #1452 can be dressed in several ways,and,after all,I think that,at least the two forms that maybe are more different between them,are different enough to classify also this bend among the bistable knots.
Below are pictures of the "Ashley form", the "Lehman form"( http://igkt.net/sm/index.php?topic=2826.msg17056#msg17056 ),and a(one of the possible)intermediate form(partially loose).
   
                                                                                                                                       Bye!

[xarax]

   In an effort to open some wide shut (mind s) eyes  , I repost the very first sentence of this thread - which is true for the "new" recently presented Plait loop, and the "old" well known Farmer s loop - as it were true for the Scot s TIB bowline, and the Ampersand bowline (1). Of course, the Plait loop is an altogether different knot from the Farmer s loop, as the Scot s TIB bowline was an altogether different knot from the Ampersand bowline -but one has to "see" that obvious fact, indeed !  ( I will say nothing about how easy is to inspect the Plait loop, in comparison to the Farmer s loop, and I will not repeat that, in the bi-stable knots, the one form DOES NOT capsize into the other...)

  The topology of a knot does not uniquely determine its geometry. Topology does determine zero to several allowable *knotted* geometrical conformations - in other words it determines a *set* of materially *possible* geometrical configurations -so it does not lead always to one and one only knot.
  So, one and one only tying diagram can lead to two topologically identical, but geometrically quite different knots...   

1. http://igkt.net/sm/index.php?topic=4877.msg31925#msg31925

[knotsaver]

I want to add the ABoK #1092 double loop to the list of the bistable knots.
Please, look at the pictures.
The first is the form shown by Ashley, the second form is obtained by tightening both ends in opposite directions.
bye,
s.