smoothly tied bends

[kd8eeh]

It has taken me some time to clarify these ideas of a "smoothly tied" bend.  I first looked at this definition to try to describe the relationship between a hugo bend and mikes bend, both of witch are smoothly tied.  I believe the concept is similar to a rope made hinge, but this definition is broadened to include almost all simple bends.  How I define a knot to be smoothly tied is:
1: The knot must be such that there is an axis about which each strand rotates in exactly one direction about this axis.
            This means that were the knot to have a pole sticking through it, each strand would wrap around this pole in only one direction, forming no cow hitch like structures not any loops that force the rope to travel opposite this direction of rotation.
2: Only the topological forms that have this property with no manupulations are smoothly tied; eg. a carrick bend is smoothly tied when it is flat, but in the form with load applied to it it is not.
3: Each strand completes at least one complete turn around the axis; ie a reef knot and a sheet bend are not smoothly tied.
When I say a homogenious knot, i mean a knot that is smoothly tied and symetric, where both tails are twisted around the knot in the same way.

This concept does not have that much value, to the extent i can find, with telling the functionality of a knot, but instead it seems to be ideal for discovering and classifying new knots.  The simplest smoothly tied knot is a zeplin knot (depending on how you define simple, but by most definitions it is).  This concept allows you to generalize on the method by which a zepplin knot is tied, and in this way i replicated a hugo bend.  Also, mike's bend is very easy to find to be homogenious, and when this is found it can be easily doubled.  Turk's heads in two strands are the core of homogenious knots.  Also, because any overhand bend can be expressed this way as a smooth knot, it gives a foundation to classify them.

I have not done any of the topology behind this definition, but i do believe there is a plethera of knots that may be described this way. 

Tell me what you think of this classification, or ask if i need to clarify this definition.

Here's a knot i made using this idea(may already exist, but it looks very nice):

[Dan_Lehman]

[ kd8eeh, you should've seen a good number of red-underscored
  words in your text; it would be a help to readers if you would
  redress the misspellings rather than click POST too soon.  FYI.
  (It won't help with 'witch' for "which", e.g., but with much.  )
]

As for this classification, it doesn't seem terribly helpful.  But
such is a tough task.  Frankly, "smoothly tied" seems an odd
moniker for what you specify.  "Axis" had me thinking of X1's
use of it to mean a physical part, but you apparently only
intend a virtual axis.  In any case, for "simple", I don't see how
the zeppelin bend is any simpler than the several other like
knots of interlocked overhands such as Ashley's #1408.


--dl*
====

[kd8eeh]

Again, this definition may be more helpful from the mathematical prospective than a practical prospective.

I do believe all overhand bends are smoothly tied, although i have yet to be able to prove it.  I think that this would enable us to find every overhand bend in existence by considering every possible way in which the two structures can be oriented, and then every possible entanglement therein. 

The reason i said a zeplin bend is arguably the simplest homogeneous knot is because the zeplin bend may be tied very easily, and consists of only one tuck in each strand.  Some others are equally simple, but less symmetric from this viewpoint (the loops are not going the same direction in some, such as the hunters bend, and in others the symmetry is flipped so that the knot is not really homogeneous. 1408 has the two strands doing mirror images of each other, not the same thing). 

The definition seems a little abstract for most things, but it explains a few phenomena very welly.  For instance, comparing mikes bend to a jar sling knot becomes easy.  In fact, i find tying mikes bend to be simpler by creating two loops around my finger, and then weaving the tails through the structure.  Also, it gives a rather simple way to tie very complicated bends, particularly for those who are turks head inclined, assuming those bends are smooth.

I would really like to get a topological opinion of this classification, because I believe it may be very significant in knot theory.  It gives the ability to wright many bends as turks heads, and this is useful mostly to topologists.  Also, it provides a framework for modifying knots, and as i have already shown, i think it will help make pretty knots.