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The relationship between 'bends' and eye knots

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agent_smith:
All 'Bends' have corresponding eye knots ('loop knots').
There are 4 possible linkages that can be made between tail(s) and S.Part(s) - from which 'eye knots' are derived (aka loop knots).
A further 4 eye knots can be derived based on loading profile where the tail and S.Part of an eye knot are swapped/exchanged.
Thus, a total of 8 possible eye knots can be derived from a 'bend'.
Not all of the derived eye knots will be necessarily stable/viable.
[Note: Linkages can only be formed between segments of opposite colour - refer to images below which employ 2 different coloured ropes to form a bend. With reference to the images, it is not possible to create an eye knot by linking 2 segments of the same colour].

All of the corresponding eye knots will share the same common core structure - thus being of the same 'genus' type.
The relationship being geometric in nature.

It appears that Harry Asher may have been the first to publish a rudimentary theory about this subject area - in his book: "The Alternative Knot Book".
However, Asher did not expand on the theory or offer any detailed insight.
Clifford Ashley and Cyrus Day don't appear to have published anything about the relationship between bends and eye knots.
I haven't found anything from Budworth.

It would be interesting to catalogue a range of end-to-end joining knots (ie 'bends) - and then derive the corresponding eye knots.
Depending on the particular bend and resulting loading profiles, some new offset bends may also be discovered with this approach.

I do seem to recall that the great knot master Xarax may have stated: "If you have tied a bend, then by definition you have also tied all of its corresponding eye knots".
Or something similar to that effect?
I am not entirely convinced of this - since some interesting eye knots emerge - with similarly interesting response to various loading profiles.

I've offered up 4 different 'bends' and their corresponding eye knots.
It would be a huge undertaking to catalog all of the known bends and their corresponding eye knots - which might be too time consuming or too difficult a task...

EDIT NOTE:
None of this information is exhaustive or the final word on the theory behind bends and their corresponding eye knots.
Its an evolving field of study - one that I think is worthy of attention.

Dan_Lehman:

--- Quote from: agent_smith on November 20, 2023, 04:30:33 PM ---All 'Bends' have corresponding eye knots ('loop knots').
There are 4 principal corresponding eye knots[/b] to any given 'parent' bend.
There are an additional 4 secondary eye knots[/b] - due to choice of 'S.Part' and tail end.

--- End quote ---
WHERE ARE YOUR DEFINITIONS OF THESE PROPOSED TERMS?!
(You also need to choose between "principal" & "primary", I think;
IMO, the latter better corresponds to "secondary".)

Let us posit that common/simple end-2-end joints & eye knots
are "2-Tangles" --i.e., as viewed so that all parts ("ends") that
exit the entanglement (the Tails, and for EKs the Eye Legs)--
then the end-2-end joint will be seen as that knot of the 2-Tangle
loading ends 1 & A in opposition to each other, Tails being 2 & B.
Eye knots are :

 on notation ::
     "1-v-2+A" means "1 is loaded in opposition to 2 & A(the eye legs )"

And I leave blank line for "ditto" repetition,
giving only as last character that of the differing
character for that line --e.g., in line 2 for "B" vice "A".


----------S.Part-loaded
1-v-2+A  ---1st
        +B  ---2nd
A-v-B+1  ---3rd
        +2 ---4th

---------TAILoaded
2-v-1+A ---5th
        +B ---6th
B-v-A+1 ---7th
        +2 ---8th

My surmise is that the first set might be called "primary"
because each has an e2e S.Part qua EK S.Part;
the lower four load an e2e joint's Tail qua EK S.Part.
But another thought is that only the 1st &3rd cases
should be "primary", as these preserve the e2e joint's
S.Parts and ADD one Tail for making an EK!?



--- Quote ---I've offered up 4 different 'bends' and their corresponding eye knots.
It would be a huge undertaking to catalog all of the known bends and their corresponding eye knots - which might be too time consuming or too difficult a task...

--- End quote ---

Your sample sets break any rule; you're apparently
operating by some sort of *feel* of propriety rather
than a logical definition.

Let's take them in presented order.
First up is the capsized Carrick Bend, for which your
"A" & "B" EKs are the traditional choices, my 1st & 3rd cases
defined above, where e2e joint S.Parts are loaded.
Then comes your "C" in which a chosen joint's S.Part
connects to the other piece's Tail; but you only give
one of these --of the TWO defined--, I guess so to keep
to your pronouncement of "four" --you got both ends
loaded for A&B but don't give a yellow-rope's play
at being S.Part (my 2nd or 4th cases).
Instead, you make your fourth be a Tail-loaded case,
but again with only one color's end qua S.Part, the
other-rope complement not counted.
Note that this case is of a symmetric knot.

Next comes the more elucidating asymmetric and
natural case (and simple!) of the net-knot 2-Tangle,
home of familiar sheet bends, BWLs, becket hitches.
You have the most commonly cited 1st case, but miss
the complementary (other rope playing S.Part)
3rd case --that, yes, awkward thing loading a U-fold
end and necessarily sending the other into a trying
backflip!  But that's logic/definition, not whim.
(It is your "C" with Tail & S.Part switched --blue Tail.)

Your third set (Butterfly) is puzzling re what the four
knots are --as the first/top case with hashed eye
connection has TWO EKs, per the blue, white options
qua S.Part (asymmetric knot, too) --and not some
new thing put in devoid introduction of "mid-line" bent!
Similarly, the bottom case also offers --as you note
by "EEL"-- two EKs, not one schizophrenic one!

Finally, to the one begetting this interesting deliberation,
with the "Lehman8", brought out by Alan's Tail-swapped
variation.  Your "B"&"C" are my 1st & 3rd, traditional
cases of one piece forming its part and then arcing
out to make the eye and return qua opposed joint S.Part
to close out; so, a yellow S.Part'd EK, and blue one too.
But then your "A" has Alan's variation --a returning yellow
eye leg into joint's blue Tail--, but not the corresponding
one for the blue eye leg fusing to a yellow Tail.  Rather,
you make your fourth case a joint's Tail-loading for EK
(for which there is the complementary yellow-Tail'd case
to do, if you fancy that.  Well, again, maybe by mere
*fancy*, but not by logic of Tangles and knot definitions.

Finally, to the assertion that tying the e2e joint means
you've tied all EKS, well, why be joint-centric in this ::
tying ANY *knot* means you've tied all of that knot's
Tangle's knots ... !?  But IMO having tried to work
with this notion of Tangle and Knots-are-loadings-of
-Tangles notion, things are tough; dressing can play
a role.  I found the Tail-Locked BWL not by plying
the fun, SIMPLE loadings for the common net-knot
(BWL, sheet bend...), but seeing the TLBwl In The Wild;
whereas in going around the logical loadings of that
Tangle, I got only something well inferior (IMO) known
as sort of Crabber's EK !!

And there is the issue of "primary" :: if your Tangle *seed*
is an EK, you have just the one S.Part, in case you have
some sort of "primary" feelings for "1" vs. "A"; with the
e2e joint, you have two.   And then each such knot can
demand full treatment for itself, where IT is the definer
of derivatives.  Consider Alan's twist on the Lehman8
qua e2e joint, e.g.!


--dl*
====

agent_smith:
Thanks for your reply Dan.

Some assumptions that I think you have made - need to be addressed.

In the first instance - my opening post (to open this new topic post) - is not exhaustive or intended to be the final word on this subject.
It seems that in your reply, you make the assumption that the theory is already well established and settled (when it is not).
In this regard, I added an edit note to my original post pointing out that this subject is not exhaustive and is very much an evolving work.

The second point I'd make is with regard to definitions (in the English language).
I have been playing with the term 'principal' and 'primary'.
I haven't settled on a best candidate descriptor...
Although given my use of 'secondary corresponding eye knots' - this does suggest 'primary corresponding eye knots' as a better candidate.

For the 'primary' derived eye knots - they adhere to the following general rules:
1. They are based on the chiral orientation of the parent bend (and here I am using the term 'parent' to refer to one specific geometric form).
2. The choice of S.Part is derived from the parent bends opposing S.Parts - preserving the loading profile of that bend (where possible to do so).
3. The primary eye knots can only be formed by linking segments of different colour (where the parent bend is tied from 2 different colour ropes).
4. There can only be 4 primary corresponding eye knots.
5. In some cases, not all of the primary corresponding eye knots will be stable/viable (it is false to declare that 100% of all corresponding eye knots will always be stable in every case).

The Xarax quote was from old memory - I do recall him making such a statement many years ago - but I have no concrete written proof.
Be that as it may, I understand the point he was trying to make.
I did make the comment that I was not entirely in agreement with him - for various reasons.

...

I am happy for you to advance your counter theories - as I think this is a subject area of significant importance.

For example, if you wish to declare a competing view that there are in fact more than 4 'primary' corresponding eye knots, I think you should take the time to back your theories with clear and unambiguous photographs (so it is clear to the viewer/reader).

I think it is open to some debate as to how 'loading profile' plays a role in the classification of corresponding eye knots.
For example, we could dispense with load altogether - and only examine corresponding eye knots without load.
This might alleviate your seeming problem with choice of S.Part and tail end.

I am in favour of preserving the general chiral orientation of the 'parent' bend.
'Primary' corresponding eye knots should (wherever possible) - follow the general loading profile of that bend, and/or be the most stable configuration.

...

As for the rest of your criticisms of each of my initial offering of 4 bends and their corresponding eye knots...
A lot of this is incoherent and appears to a source of irritation on my depicted choice of 'S.Part' and 'tail end'.
Again, I have tried to preserve the chiral orientation of the parent bend - and to use the bends existing 'S.Parts' (where possible).

Your seeming irritation of choice of 'S.Part' and 'tail end' could be somewhat alleviated by dispensing with loading profile and only looking at the geometry of eye knot.
This is something that needs further thought...

In any case, I'll keep posting while I am motivated to do so.

My next offerings will be the infamous 'Riggers bend', the Zeppelin bend, and the Figure 8 bend (#1411)..
The derived corresponding eye knots from these bends are interesting.

Amen.

Dan_Lehman:

--- Quote from: agent_smith on November 21, 2023, 03:55:37 AM ---It seems that in your reply, you make the assumption that
 the theory is already well established and settled (when it is not).

--- End quote ---
Not at all; I pointed to the glaring absence of statement
of your theory --i.p., how do you define your terms.
To which purpose I laid out the logical possibilities for
a 2-Tangle's 8 eye knots ::

----------S.Part-loaded
1-v-2+A  ---1st
        +B  ---2nd
A-v-B+1  ---3rd
        +2 ---4th

---------TAILoaded
2-v-1+A ---5th
        +B ---6th
B-v-A+1 ---7th
        +2 ---8th

Now, it is for YOU to define your "primary/secondary" definitions
using these articulated cases.  I showed how your offered sets
of supposed primary knots followed no rule, but some whim.

Now, you give this --which fails to satisfy--:

--- Quote ---For the 'primary' derived eye knots - they adhere to the following general rules:
1. They are based on the chiral orientation of the parent bend

--- End quote ---
This tells us nothing; and "chiral orientation" is unwanted.


--- Quote ---2. The choice of S.Part is derived from the parent bend's opposing S.Parts
 - preserving the loading profile of that bend (where possible to do so).

--- End quote ---
Again, "is derived from" says nothing; it begs the question HOW?
(One might state, e.g., "One of the joint's S.Parts will be the EK's S.Part.)


--- Quote ---3. The primary eye knots can only be formed by linking segments of different colour ...

--- End quote ---
This gratuitous and unwantedly incorporating a presentation aspect
(nicely contrasting colored ropes) : an EK is just that, and so clearly
not a link.

--- Quote ---4. There can only be 4 primary corresponding eye knots.

--- End quote ---
!!??? Whosever ordained this limit?
Rather, one might guess/hope that this statement is merely
a plain consequence of the yet-to-be-given definitions of
primary/secondary EKs of a joint.  (And beware the case(s)
where in fact there are not so many distinct knots --albeit
yes the filling out of my EK possibilities which number 8 :
there might be duplicates.



--- Quote ---For example, if you wish to declare a competing view that there
are in fact more than 4 'primary' corresponding eye knots,
I think you should take the time to back your theories with
clear and unambiguous photographs (so it is clear to the viewer/reader).

--- End quote ---
Photographs --of logical entities?
Their representation is perspicuously given now twice;
ANYone can attach the "end" labels and see the results
of treating some knot in Tangle analysis.


--- Quote ---I think it is open to some debate as to how 'loading profile' plays
a role in the classification of corresponding eye knots.
For example, we could dispense with load altogether
--and only examine corresponding eye knots without load.
This might alleviate your seeming problem with choice of S.Part and tail end.

--- End quote ---
?????  This is nonsense :: the EKs are defined by loading profile
of a given Tangle --as e.g. in the BWL 1010 & 1034.5 from the
loading of that common 2-tangle.  The loading profile on the
Tangle is how EKs are defined!


--- Quote ---I am in favour of preserving the general chiral orientation of the 'parent' bend.
'Primary' corresponding eye knots should (wherever possible) follow the general
loading profile of that bend, and/or be the most stable configuration.

--- End quote ---
Well, this shows you have only a *feel* and whim about
what you will choose to accord the gold star of "primary";
that said,  I don't understand how you come to think there're
only four --your sets, as I've shown, have in them knots
that have a complementary (colors reversed) equally
qualifying knot that you've left out.


--- Quote ---As for the rest of your criticisms of each of my initial offering of 4 bends and their corresponding eye knots...
A lot of this is incoherent and appears to a source of irritation on my depicted choice of 'S.Part' and 'tail end'.
Your seeming irritation of choice of 'S.Part' and 'tail end' could be somewhat alleviated
by dispensing with loading profile and only looking at the geometry of eye knot.

--- End quote ---
  It is not at all incoherent but carefully stated,
and your choice of S.Part/Tail is exactly what defines a knot.


--- Quote ---My next offerings will be the infamous 'Rigger's bend', the Zeppelin bend,
and the Figure 8 bend (#1411)..
The derived corresponding eye knots from these bends are interesting.

--- End quote ---

No, please not.  You stand in want of a definition that can be applied
by people to understand things; they are not waiting to see how YOU
understand things one whim to the next.  (Or even if that, how it is
you can assert that four is the limit.)


--dl*
====

agent_smith:
In reply to Dan,

I read your post(s)... And after sorting through the negativity and generally incoherence;
its hard to avoid reaching a conclusion that there is something 'off' here.
Your closing remark for instance:

--- Quote ---No, please not.  You stand in want of a definition that can be applied
by people to understand things; they are not waiting to see how YOU
understand things one whim to the next.  (Or even if that, how it is
you can assert that four is the limit.)
--- End quote ---
I am not sure if this is in the spirit of this forum?
Okay, I could just simply yield and stop posting any further bends and derived corresponding eye knots.
But is that in the best interests of exploring this subject?
Should I just yield and leave it to you to further this on your own?
I'll point out (again) that this entire subject area is new and explorative - it is certainly not settled.
There are no existing research papers or peer reviewed works that I can draw from.
I am forging my own path - where none exists (so to speak).


--- Quote ---yes the filling out of my EK possibilities which number 8
--- End quote ---
Hmmm, I see you have offered up a number!
8, it seems :)
(I also had previously pointed this number out - which was met with debridement - but now is okay if you arrive at the same number?).

Now I could serve up the same sort of narrative back to you.
To para-phrase you: 
--- Quote ---!!??? Whosever ordained this limit?
--- End quote ---

Can you prove that there is such a limit?  ;)

I wont go on here because it would serve no useful purpose.
Other than to say that I still posit the following general principles:
1. For any given 'bend' - there will be corresponding eye knots. EDIT: There are 4 possible linkages - from which a corresponding eye knot is derived.
2. There will be a maximum of 2 'primary' corresponding eye knots and up to 6 'secondary' corresponding eye knots (a total of 8).
3. This 'correspondence' is based on the fixed chirality and geometry of the parent bend (ie the 'chiral form') - a different bend means another set of different eye knots.
4. The 'primary' eye knots are closest to the parent bend in terms of loading profile.
5. A further 4 eye knots can be derived based on choice of S.Part (the tail and S.Part are swapped) - therefore a total of 8 possible eye knots can be derived from a parent bend (but not all of which will necessarily be stable/viable).

These statements are not settled - that is, it is not an established/agreed theory.
This is a relatively new approach to understanding the link between a given 'bend' and its 'corresponding' eye knots.

The source of irritation with Dan appears to be linked to:
1. What is the definition of a 'primary' corresponding eye knot?; and
2. What is the method be which an 'S.Part' and 'tail end' is chosen? (to distinguish it as being 'primary').
3. The total number of possible corresponding eye knots - 8 - may be in dispute?
4. The gratuitous use of two different coloured ropes to depict a parent bend.

Its a pity that the source of irritation and shouting cant be managed in a more civilised manner.
When you think about it, there are no power struggles here, or attempts to dominate, or to vanquish an opponent.
Its just an attempt to try to open some new lines of thinking - which may have been started way back with Harry Asher's book (1989?).

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