The continuing conversation about 'correspondence' between knots (not tangles).
It is difficult to visualise any correspondence between a tangled mess relative to another tangled mess.
Only you keep imputing this silly notion to satisfy your
I-don't-know-what; "Tangle" is defined for the purposes
of my articulation w/o any notion of "mess", but of pure
*entanglement* of material, various parts between/around/
against each other, and being defined knots by applying
a Loading Profile. --a seemingly attractive PoV, until, alas,
the hard nitty gritty sets in, and there come some leakage
of that hoped-for articulation of knots. <sigh>
My definition of a tangle being:
An unintentional confused mass of twisted cord/rope - that has no ...
... bearing on MY use of the defined term per MY
system of articulation.
My broad definition of a knot:
... tied structure with a distinct geometric form ...
The problematic aspect of "intention" & "recognition"
have been discussed elsewhere; here, let's lean on
"tied structure" & "with a distinctive geometric form" ::
Knots well known & used can run through some range
of geometric shapes, as I previously noted with the case
of the BWL's range of helix angle in the central nipping
loop --getting the the point where one really doesn't
want to call it a "loop" anymore, but at what helix angle
is this to be decided --which angle, btw, will likely change
per force upon the knot!
.:. This is a tough issue!
My definition of a 'bend':
An end-to-end join of cord/rope/webbing (ie textile material).
There are 2 ends, and they are united to enable through loading.
Noting that "bend" from the nautical-cordage-using
world has a different meaning --why press Ashley's
lean to ignore this, and lose the connection to the past,
vs. going with --as you have nicely offered/done, too--
"join(t)" --though one might think a hitch also joins ...,
and ...!? (I find "join" irritating, also as I find "tie" which
some writers use.)
My definition of a 'hitch':
A structure that requires a 'host' to form around and function under load.
Somehow you're getting the cart before the horse,
rather than making your "structure" at once :: i.e.,
there is this hitch thing looking around for its host;
to me, rather, it is a knot in which cordage is tied
to some object. And I'll avoid the superfluous note
that --true or not-- removing the host does ... <whatever> ::
the "hitch" has this "host"/"object", and what happens without
that is just out of discussion. (kinda like saying that if you
remove the other end of a joint then the joint will lose ...)
My definition of a 'knot'
(taking into consideration the broader definition of what a knot is):
A self-supporting structure tied from cord/rope.
No 'host' is required, and there is no union of two ends.
So, if you come across a Butterfly Knot you'll say
"this knot vanishes when I cut its eye!"
I leave "knot" as the for-everything term,
which includes hitches, binders, et cetera.
With regard to correspondence:
All 'bends' have 4 possible linkages that can be made between the S.Parts and tail ends.
These linkages create eye knots.
A further subset of 4 eye knots can be created by way of transposition of S.Part and tail end.
Not all of these structures will necessarily be stable/viable.
I think you want that warning re stability to cover the
whole 8-items lot --think of the SquaReef Joint, e.g.;
but then wonder what is meant by working/not-working,
in light of the variety of *cordage* (my catch-all for
"knottable material", which I've otherwise called
"Piece of Flexible Material" to generalize).
Definition of transposition:
For transposition of eye knots:
Where an S.Part is exchanged with a tail end but without altering the knot core geometry.
Transposition can be likened to a reversal of polarity.
Again, I think that the Tangle+LoadProfile=Knot articulation
gets here more quickly, obviously, understandably, and without
having to invoke another concept :: one doesn't exchange
anything, but rather uses a different loading profile
(which sort of alteration could occur in practice).
Definition of 'core':
The part of a knot that is central to its existence or character.
This begs the question "What is it's "character"?!
I'd like your example of what sort of knot has a core
that is not what Chisholm calls its "nub" and what the
Tangle articulation sees as the tangle --though how
to treat the Sheepshank and the shared-eye joints
is baffling me so far <argh>! E.g., the Twin BWLs
of Ashley, where each BWL's Outgoing Eye Leg
is the Ret.EyeLeg to the opposite BWL ! How
to treat this "shared eye" span of unentangled
cordage (like that in a Sheepshank --which also
challenges how to regard the nipped u-folds that
lie beside each S.Part!?
Definition of 'eye knot':
A self-supporting structure that has a connective eye that is non-slipping.
The eye has no chirality.
"Self-supporting" seems superfluous, perhaps informative?
"connective eye" begs for definition. "non-slipping" hints
at troubles to encounter with vagaries of materials & forces!
(How do you regard the fabled --we might have to rely on fable--
Tarbuck Knot? --intended to slip BUT only so much, then not.
There is no dispute that if you take the joint as a 2-Tangle
then there are 8 per definition EKs (possibly with duplicates).
Except, alas <sigh>, I found that the purely logical direction
of a Tangle + Loading_Profile doesn't ensure that there's not
variation in this specification --that going from the Tangle
to this state by simple loading these "ends" just so! can
run afoul of practical consequences ("Just how does one get
to just so?!").
My approach is to constrain 'correspondence' in terms of a geometric relationship, rather than the result of loading.
You may prefer to choose a different approach.
Rather than result it is simply THERE, logically;
actual loading might lessen one's value for the logic, though.
But there is another way to see "correspondence" of an EK
to a joint. I remarked about Ashley's #1043/5 --and I should
add #1016 (or some version of...)-- being seen as corresponding
to the Sheet Bend
#1016 is topologically equivalent to #1047 Bowline with a bight:
I think you're missing my point :: that one seeks to find
correspondence and preservation of the loading
via tying an end(1)-to-endS(2 --of the U-fold eye) joint
AND THEN somehow fusing one of the eye-leg ends
into some part of the S.Part's flow. The example above
re the angler's Blood Knot is of a very good one; that
of the #1043/5 cases illustrates the concept also nicely;
whereas trying to make a Fig.8 joint=>EK this way ...
simply looks absurd vs. the usual Fig.8 EK.
...
"a good example":: well, in fact what brought this idea to
mind, for me, reading of the "2-by1 knot"[?! --name not sure]
in Stanley BARNES's 1951 ed. Angler's Knots ::
...
This makes it difficult for readers to follow your ideas.
But readers should know the Blood Knot of anglers,
and otherwise can find it. To that, one need only
follow the words and form the joint 1 line-vs-2.
The version of the Blood knot that works simplest
would take the two ends' Tails and fuse them;
then you cut off one S.Part to amount of material
for fold back to form the eye,
and run it back twin to its extant course.
(In practice, one will I think want fewer of the
binding wraps from EACH eye leg, summing to
about the number in the S.Part's wrapping.
**MY insight to this revelation was to see that one could
*fuse* together one of the eye-part's Tails to the S.Part's
Tail and so render the structure in a nearly identical
single-line Eye Knot.
Again, hard to visualise your revelation without a reference image.
Fusing one 'eye parts tails' to the S.Parts tail sounds like a revelation... but
very difficult to provide any meaningful comment.
The reference is the Blood Knot; tying 2-to-1 will see a trio
of Tails in the nipped center; now, the task is to make a fusing
of them to render a single-cordage piece Eye Knot.
--dl*
====
