Constrictor, Strangle and Transom : the "same" knot, or not ?

[xarax]

  Two or more knots can be topologically equivalent, yet geometrically very different - and, because it is geometry what determines how a knot "works", they should not be considered as the "same" knot. A few, only, examples of the many such "bistable" knots that exist, are shown at (1).
  However, I believe that the exact opposite may be also true... Two or more knots can be geometrically very similar, and so they can "work" in a similar way, yet they can be topologically different. To consider that two topologically different knots are, in a sense, the "same" knot, is almost a blasphemy to a (mathematical ) knot theorist, where topology, and topology only, rules - but not to a (practical) knot tyer. On the contrary, for a knot tyer this attitude is a comprehensive and useful way to "see", with his mind s eyes, knots which are "dressed" similarly, but which are also "dressed up" like deceptively different knots, and are called by different names, as essentially similar knotting mechanisms, as variations of the "same" knot.   
  An example of this are the three "different" well-known, and great knots, the Strangle, the Constrictor and the Transom - as one can see at a glance, by looking at the attached picture. Who can offer convincing arguments that the two different ways the four limbs can be connected at the back side of the hitched/bound object(s) and form two wraps, make the knotting mechanism to "work" differently, and those two configurations/variations to become two "different" knots ? ( The reader is not advised to bet on which knot he sees... ). The heart of the knotting mechanism which "locks" and secures the two ends of those tight hitches lies at the "front" side, at the central area where the lines are twisted around each other, so the exact way the lines travel at the "rear" side of the object(s) is mechanically irrelevant.
   ( I admit that this was also a cunning little game to increase the capacity of my knotting toolbox, without increasing the maximum, total number of the "different" knots it should contain  - but that is another matter (2))

1. http://igkt.net/sm/index.php?topic=4201   
2. http://igkt.net/sm/index.php?topic=1150.msg33735#msg33735

[xarax]

  If the diameter of "front" hitched / bound cylinder-shaped object becomes larger, we tend to "see" a Strangle. If it becomes smaller, we tend to "see" a Constrictor. As the diameter(s) of the hitched / bound object(s) should not determine the kind of the hitching / binding knot itself, we conclude that both knots are implementations of the same knotting mechanism, therefore they are essentially the "same" knot.
   ( If we split this hair a little more, we can say that a substantial change of the diameters of the hitched / bound objects also means a substantial change of the geometries of the hitching / binding knots, and since geometry is what determines how a knot "works", different geometries mean different knots, etc...)     

[Dan_Lehman]

If the diameter of "front" hitched / bound cylinder-shaped object becomes larger,
we tend to "see" a Strangle; if it becomes smaller, we tend to "see" a Constrictor.

I'm not sure I concur in this,
but either way, if the apparent front object is seen newly
as the rear --easily done in the lower image--, then the actual
knot necessarily changes from one to the other.

IMO, when used in this "transom" binding, the two distinct
binders swap characteristics --the better binding becomes
less so, as the effective geometries of nipping in relation to the
object are switched around !!  --very interesting.  (My initial
reaction to Ashley's assertion @#385 of the transom knot being
"a modification of the constrictor" was that he was mistaken;
but I see now that there is merit to this, despite the knot alone
being a strangle (it *strangles* the away object, but *constricts*
against the near one).

Two or more knots can be geometrically very similar, and so they can "work" in a similar way,
yet they can be topologically different.

Here, though, one must realize that the geometry of curvature
significantly differs : for a (canonical) strangle the ends run
parallel with the round-cross-secional object, vs. those of the
constrictor running perpendicularly.  In the *transom* binding,
there is essentially a like object inserted perpendicularly to the
knot-defining object between the area of entanglement and the
original object, thus switching the physical mechanim via this
change to the immediate surface geometry.

As the diameter(s) of the hitched / bound object(s)
should not determine the kind of the hitching / binding knot itself,
we conclude that both knots are implementations of the same knotting mechanism,
therefore they are essentially the "same" knot.

Or we don't, and conclude something more puzzling!
.:. a fascination worth indulging, whatever!            

--dl*
====

[xarax]

...if the apparent front object is seen newly as the rear --easily done in the lower image--, then the actual
knot necessarily changes from one to the other.

   Actually both poles have the same diameter, it is only the perspective distortion of the almost macro shooting that gives this impression. I had only wanted to point out that appearances may be misleading ! 
   Anyway, you got the idea - when we change the diameters of the hitched/bound objects, we change the names of the hitching/binding knots ! Sympathetic magic ! 
 
   

   Here, though, one must realize that the geometry of curvature significantly differs : for a (canonical) strangle the ends run parallel with the round-cross-sectional object, vs. those of the constrictor running perpendicularly.

   Yes, but only in relation to the object ! In relation to the knotting mechanism itself, they do not. "Parallel" and "perpendicular" orientations can only be defined in relation to the "environment" of the knot, not in relation to the knot "per se". This is the point : The knot changes, not because of intrinsic changes, but because of its environment. So, the knot changes in quality ( kind / species ), when the environment changes in quantity ( size ).

   The other way out, is to ( be forced to ) include the object "in" the knot, to consider the object as part of the knotting mechanism, and of the knot itself. Although this will lead to a proliferation of the number of "knots", ( which is exactly the opposite of what I was trying to achieve in the first place  ), it can easily be done when the object is a segment of flexible, compressible rope - but when it is a piece of a rigid, incompressible pole ? ? ? 

[Tex]

I would lash the heck out of it and quit worrying about it. 

But I see a strangler because the near object defines the knot curvature. 

The word Palomar comes to mind.  Without the object, it's just a half hitch on a bite.  The word defines not only the knot geometry, but yes also the object, and how to apply the knot to the object.  Probably from a theoretical perspective this does not warrant a new name, but if you're fishing it does.

[Tex]

Without any kind of object I meant to call it an overhand on a bight of course, but there you go.

[xarax]

   the near object defines the knot curvature. 
   Without the object

   Without any kind of object

  I was not talking about the presence or the absence of the hitched/bound object(s) ( without those objects, obviously the hitches become stoppers, etc...), but about their size. By changing the relative sizes / diameters of the two objects / poles, the one knot is transformed into the other. Anyway, I think that if one sees the two knots from this perspective, he will tend to distinguish more what are their common, constant, essential characteristics, and less what are the different, varying, secondary ones. The knotting mechanism is the same, the sizes of the hitched/bound objects and the names and orientations of the ends of the hitching/binding knots are different.