The relationship between 'bends' and eye knots

[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]

@agent_smith post:1

All ‘Bends’ have corresponding eye knots (‘loop knots’).
There are 4 [b]principal corresponding eye knots[/b] to any given ‘parent’ bend.
There are an additional 4 [b]secondary eye knots[/b] - due to choice of ‘S.Part’ and tail end.

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!?

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...

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]

@agent_smith post:3

It seems that in your reply, you make the assumption that
the theory is already well established and settled (when it is not).

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–:

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

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

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).

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.)

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

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

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

!!??? 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.

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).

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.

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.

????? 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!

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.

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.

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.

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

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.

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:

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.)

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).

yes the filling out of my EK possibilities which number 8

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:

!!??? Whosever ordained this limit?

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?).

[Dan_Lehman]

@agent_smith post:5

… the negativity and generally incoherence;

Let’s see exactly what you find of generally incoherence?
My words are there in hard print for all to see, to refer to
exactly, not in general. Perhaps I might clarify something for you?

Your closing remark for instance:

You stand in want of [u]a definition that can be applied by people[/u] to understand things; ...

Pretend that you're too busy to treat those threatened further cases of joints begetting eye knots, and so you solicit helpers :: now, how do you instruct those helpers to examine the joints and show what you call "primary" (& secondary) corresponding eye knots for the joint? THIS instruction, the definitions that must guide any to follow in your discriminations (at least, if they are to be generally followed), is what you continue to lack.

yes the filling out of my EK possibilities which number 8

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?).

"debridement"?? The number is quite evident in the definition given of a 2-Tangle, but that only. I have also noted that there are other means of making what might be called a "corresponding eye knot (to a joint)". But, of a 2-Tangle, there are 8 EK loading profiles, by definition.

Now I could serve up the same sort of narrative back to you. To para-phrase you:

[i][b]!!??? Whosever ordained this limit?[/b][/i]

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

But that is no limit, but mere fact of the 2-Tangle --four ends to be an EK S.Part, and two loadings for each in this. (The count of *knot*s as unique is another matter :: e.g., in the [i]Carrick Bend's[/i] case, you showed yellow & blue S.Parts each making the same knot --the obvious, traditional one.)

But YOU have not given any hint of a reasoning for your asserted
limit of four --nothing; and your sets show varied inclusions/exclusions,
some of which are clearly illogical (a yellow S.Part in, but the matching
blue S.Part case not, e.g.). Now, you seem to want to feel around
for behavioral niceness in order for you to award some EK “primary”
status; well, accepting that that’s one way to look at knots, it begs
the question of Why (only/as many as) four?! To which you’ve had
no response --coherent or other wise.

I won't go on here because it would serve no useful purpose. [u]Other than to say that I still posit the following general principles:[/u] 1. For any given 'bend' - there will be [i]corresponding [/i]eye knots. 2. There will be 4 'primary' corresponding eye knots and 4 'secondary' corresponding eye knots (a total of 8).

HERE YOU GO AGAIN :: WHY, OH WHY, ARE THERE >>4<< "[i]Primary[/i]" EKs?? 1) What makes an EK [i]primary [/i](for SOME e2e joint)??

3. This 'correspondence' is based on the fixed chirality and geometry of the [i]parent bend[/i] (ie the 'chiral form') - a different bend means another set of [i]different [/i]eye knots.

What is THE chirality of an e2e joint in which one piece's handedness is Z but the other's S?! (E.g., the shared-eye joint known sometimes as Two BWLs, which give a sort of "mirrored BWL" look courtesy of the two hands!?

In any case, I don’t see the relevance of this #3 criterion,
in that one has A specified joint --whatever it is, however
you name it-- and that is that from which corresponding
EKs are sought.

4. The [b]'primary' eye knots[/b] preserve the general loading profile and orientation [i]of the parent bend[/i], and are the most stable forms - where possible to do so,

Here I see you as mixing guidelines/rules :: your initial "preserve ..." suggests that one need examine each of the 8 logically possible 2-Tangle EKs (this particular treatment, not including other ways of seeing "correspondence") for behavioral nicety. Ok, that might be a way some want to go, especially from a pragmatic standpoint; but then I don't see how the number four is reasoned --which smacks of the sort of 2-Tangle figuring that I laid out, and not the process of examination for behavior (or "whim", as I've remarked)!? Perhaps I could understand four as a [u]limit[/u], but not as an assured [i]There WILL Be These[/i] set, come hell or high water: one gets such assurances by definitional implications, not whim.

5. The 'secondary' eye knots arise from the fact that for each corresponding eye knot, there is a choice of 2 'S.Parts'.

NO :: for a (simple, 2-Tangle) joint there are four ends; same for an EK. For each of the joint's ends when taken to be an EK S.Part the Outgoing Eye Leg is fixed (it is the other end of that piece giving the S.Part) can return as the joint's yet-to-be-assigned loading-- S.Part or Tail. For the common case of the Sheet Bend, the turNip has the S.Part as one end and the other is the Outgoing Eye Leg, which can take either the Sheet Bend's other S.Part (=>#1010) or the Tail (=>#1034.5). And so on. But we aren't choosing such options from an EK.

[u]The source of irritation with Dan appears to be linked to:[/u] 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').

BINGO! And alllll of these many words later we remain wanting these!

3. The total number of possible corresponding eye knots - 8 - may be in dispute?

There is no dispute that if you take the joint as a 2-Tangle then there are 8 per definition EKs (possibly with duplicates).

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; in a way, they do so in the spirit I take of your
looking for preserving loading profile and so on --which they do
by altering the reference joint from a 2-Tangle to a 3-Tangle
in which one piece forms the eye and presents thus its two
ends qua single end to fill the path of the joint. Having done
this, then look to how to fuse the single-S.Part’d piece’s
Tail into one of the twinned pieces’ Tails to restore a 2-Tangle.

4. The [i]gratuitous [/i]use of two different coloured ropes to depict a parent bend.

Entirely missing the complaint :: the use of colored ropes is GREAT, but that aspect should NOT be [u]part of your definitions[/u]. (One might print out such things in B&W and still there should be clarity and the [i]rules of correspondence[/i] need no altering.)

It's a pity that the source of irritation and shouting can't 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?).

I'm sorry that you feel that way about strongly expressed arguments, but my expression reflects my own struggles at coming to ways (even one!) to better understand knotting, and your continued failure to answer clear questions aggravates that frustration. As I said above, we want from you your rationales for the otherwise seemingly arbitrary, even capricious assertions of how things are.

–dl*

[agent_smith]

I’ve evolved the underlying theory for this subject matter.

I decided to use excessive gratuitous use of coloured dots to identify the various segments of a corresponding eye knot relative to its parent bend.
This enabled me to devise a stricter definition of what constitutes a ‘primary’ corresponding eye knot.

The primary corresponding eye knots have a congruent S.Part + one congruent eye leg.
That is:
The S.Part of the primary corresponding eye knot is congruent to an S.Part of the parent bend
One eye leg of the primary corresponding eye knot is congruent to the opposite S.Part of the parent bend.
(for now that’s as clear-cut as I can be until I examine this subject further).
The primary corresponding eye knots are therefore closest to the parent bend in terms of loading profile.
As such, I now surmise that loading profile is the best way to examine the corresponding eye knots - and to determine congruence with the parent bend.

Using this approach, it appears that there is a maximum of 2 ‘primary’ corresponding eye knots.
The remaining corresponding eye knots can be 6 or 7 in number.
Interestingly, the Sheet bend only has 1 primary corresponding eye knot.

Feel free to shout and launch a counter-offensive theory

---

[Dan_Lehman]

@agent_smith post:7

I decided to use excessive gratuitous use of coloured dot

Good! (-;

Using this approach, it appears that [b]there is a [u][i]maximum [/i][/u]of 2 '[i]primary' corresponding eye knots[/i][/b].

What might see the limit not reached (i.e., there being but 1 PCEK)?

The remaining corresponding eye knots can be 6 or 7 in number. Interestingly, the Sheet bend only has 1 primary corresponding eye knot.

This implies that the answer to my question above is one of judging the resulting knot's "stability" or the like --the EK not rising to the level of decent behavior. (The same-side Sheet bend yields a sort of Gnat-H Noose which CAN behave qua EK, in some materials, at some forces --a YMMV situation.)

Hmmm, one can wonder if there is some knot that will behave
erratically in either “primary” loading!?

–dl*

[agent_smith]

I have uploaded the corresponding eye knots for the following ‘bends’:
Riggers bend (#1425A)
Zeppelin bend

The Zeppelin bend has 2 ‘primary’ corresponding eye knots.
Interestingly, the Riggers bend only has one primary corresponding eye knot (due to its symmetry).
The geometry of the primary corresponding eye knots derived from the Riggers bend are in fact identical.

…

I looked at the Reef knot (bend) #1402 , however, there are no stable corresponding eye knots.
This is due to its structure, which is the integration of 2 opposing ‘bights’ - which are essentially 2 U folds.
Any attempt to create a corresponding eye knot results in instability.

Interestingly, a ‘Granny knot’ (#1405) does enable somewhat stable corresponding eye knots to be created.

The Ring bend (water knot / tape knot) #1412 does have stable corresponding eye knots.

EDIT NOTE:

Tugboat ‘B’ bend and corresponding knots added.
This bend had no apparent name - other than the eye knot published by Dave Root (although I have no knowledge as to actual discovery and creator info).

There is also a Tugboat ‘A’ bend - although it does not appear to be as stable as the ‘B’ version.
The difference being a mirror reversal of the Crossing hitch.

---

[agent_smith]

#1412 Ring bend (Water knot/bend, Tape knot, Overhand bend) and corresponding eye knots.
This ‘bend’ consists of 2 integrated Overhand knots of the same chirality.

I am only showing the 4 principal ways of linking tail to S.Part based on S/S chirality.
Obviously, the 'Ring bend can also have 4 linkages based on Z/Z chirality.
Linkages can only be formed between segments of opposite colour (blue to white).

These ‘linkages’ all form corresponding ‘eye knots’ (aka loop knots).
Of these, only one (1) is a ‘primary’ corresponding eye knot (2 linkages are in fact identical).
The rest are all ‘secondary’ corresponding eye knots.

Note:
This is all a new area of rapidly evolving concepts.
The terms ‘linkages’, ‘primary corresponding eye knots’, ‘secondary corresponding eye knots’ are my own.
Harry Asher likely originated the correspondence between a bend and eye knots in his book “The Alternative Knot Book” (1989).
However, he did not expand on his work or offer detailed insight to the underlying theory.

---

[Dan_Lehman]

@agent_smith post:10

#1412 Ring bend (Water knot/bend, Tape knot, Overhand bend) and corresponding eye knots.
This ‘bend’ consists of 2 integrated Overhand knots of the same chirality.

I am only showing the 4 principal ways of linking tail to S.Part based on S/S chirality.
Obviously, the 'Ring bend can also have 4 linkages based on Z/Z chirality.

And in general consideration, mirror images are the SAME knot
–you don’t get New Knot Awards for inventing such a knot, e.g.!

Of these, only one (1) is a 'primary' corresponding eye knot (2 linkages are in fact identical)

And so too are there 2 linkages (of the 8 logically coming for a 2-Tangle) in fact identical for each of the possible other loadings.

You’d do well to observer the Tangle ID scheme of labelling
each end of each entangled piece respectively “1” & “2”,
“A” & “B” (and for a 3-Tangle, “X” & “Y” --which letters I
think will be better accepted/perceived-as-different-pair
than say using Roman or Greek numerals).
As shown above, if one is specifically looking to find EKs
out of a end joint’s 2-Tangle, you run through each of the
four ends (1,2,A,B) playing EK S.Part and see what comes.

----------S.Part-loaded [here, parens added of ends joined in loading “v” other(s)]

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

In your graphic, you wrongly assert that in addition to
the two-which-are-identical primary EKs (resp. 1 & A qua S.Part)
there are “7 secondary eye knots” --wrongly merging
the two equal-result loadings (different material/color
if you tie it, but same result/knot) as “1” and subtract
from logical 8 … which misses the point that your 1
=2 loadings and so too will those other loadings have
“identical” (but for colors) knots! Half of 8 = 4 -1 =3.

You’ve missed the case of loading the EXTERIOR twin
–Tails-loading, here. (E.g., your image A but with
white Tail (upper end) qua S.Part.)

[u]Note:[/u] ... The terms 'linkages', 'primary corresponding eye knots', 'secondary corresponding eye knots' are my own.

And I can see reasons to abandon them, in time :: they have *quality*/*value* connotations, and it will probably be better to be more objective --e.g., as I have come by abandoning "strong/weak form" (given w/o good basis, even) of twin-parts knots such as this & [i]Fig.8[/i] for "exterior/interior", resp. of the loading on twinned parts.

The Butterfly’s established/defined EK loading is not
that given for e2e joints per our thinking here; it is
either 1-v-(2+B) or A-v-(B+2) --the case with both
joint S.Parts loaded (one the Outgoing Eye Leg) is
abnormal tying (cutting away the B.'s OELeg will
yield a novel joint --or two, as it’s an asymmetric
tangle!).

–dl*

[agent_smith]

Dan,

You’re making some assertions here with : you typed, “wrongly assert”…
I think we need to set the facts straight and sort out definitions.
I’ve attached a graphical representation of an 'end-to-end joining knot (‘bend’).

My first point:
With respect to any bend, there can only be 4 principal ‘linkages’ between tails and S.Parts segments (refer to the graphical representation of the a bend below).
Syntax tails…S.Parts… protruding segments… whatever!
Note the use of the term ‘linkages’.
Note also the use of the term ‘principal’.
A linkage can only occur between segments of opposite colour.

Now you either agree with that premise - or you don’t.
If you disagree, I’d like to see your evidence to the contrary.

My second point:
Each of these 4 principal linkages will form corresponding eye knots.
That is, each eye knot corresponds to its ‘parent’ bend.
Each of these corresponding eye knots has 2 possible choice of S.Parts - which makes a total of 8 possible combinations (if we think in terms of loading profile).

My third point:
Of the corresponding eye knots, some will be ‘primary’ and some will be ‘secondary’.
Definition of ‘primary corresponding eye knot’:
The primary corresponding eye knots are closest to the parent bend in terms of loading profile.
The S.Part of a primary corresponding eye knot is congruent with the S.Part of the parent bend.
One leg of the eye is congruent with the opposite S.Part of the parent bend.

…

With respect to #1412 Ring bend and its corresponding eye knots…
There are 2 linkages which create ‘primary’ corresponding eye knots.
It turns out that both of these eye knots are in fact identical.
As such, I count that as one (1) - not 2.

Now, if you swap/exchange the S.Parts of these particular primary corresponding eye knots, they are no longer ‘primary’.
(refer to my definition of ‘primary’).

And once again - in case its not clear…
I am forging a new trail here - where none exists.
I have had to evolve my own definitions and underlying theoretical concepts.
You may disagree with these concepts - and that is how science is done.
Someone publishes a paper and others will either refute it or find that their test/analysis results support it.

If you have an opposing theory - I’d like to try to understand it and your supporting definitions.

EDIT NOTE:
With respect to your original hypothesis of ‘strong’ versus ‘weak’ forms of #1047 F8:
Strength is irrelevant.
There is no load that a single climber/abseiler/canyoner/caver can generate that will reach the MBS yield point of a knot.
So you are ‘correct’ to abandon this origiunal theory.

In reality, what needs to be brought to the forefront of thinking is jam resistance, and stability.
A sloppily tied #1047 F8 will rapidly deform (refer definition of ‘stability’).
A well dressed F8 tends to maintain its geometry and does not deform to any significant degree.
The jamming threshold will be higher in one particular dressing state than the other (which corresponds to your original ‘strong/weak’ assertion).
Not much testing has been done - there needs to be some peer reviewed studies/tests (with a ‘control’ group) to settle the mater.

---

[Dan_Lehman]

@agent_smith post:12

You’re making some assertions here with : you typed, “wrongly assert”…

And one is right on target.

I think we need to set the facts straight and sort out definitions. I've attached a graphical representation of an 'end-to-end joining knot ('bend').

My first point:
[i]With respect to any bend, there can only be 4 principal ‘linkages’
between a tail and an S.Part segment
(refer to the graphical representation of the a bend below).

Wrong per your image :: you have a linkage between Tails;
between “S.Parts & Tail” we might reject the S.P-2-S.P linkage, too
–4-2 = 2.

Note also the use of the term 'principal'. A linkage can only occur between segments of [u]opposite [/u]colour.[/i]

Oh, indeed I noticed "principle" appearing on the stage!

If you disagree, I'd like to see your evidence to the contrary.

--just cited.

[u]My second point:[/u] Each of these 4 principal linkages will form corresponding eye knots. That is, each eye knot [i]corresponds [/i]to its 'parent' bend. Each of these corresponding eye knots has 2 possible choice of S.Parts --which makes a total of 8 possible combinations (if we think in terms of loading profile).

--agreed.

Definition of 'primary corresponding eye knot': [b]The primary corresponding eye knots are closest to the parent bend in terms of loading profile.[/b] The S.Part of a primary corresponding eye knot is congruent with the S.Part of the parent bend. One leg of the eye is congruent with the [i]opposite [/i]S.Part of the parent bend.

This definition should cut to the chase, and just give the loading profile --as written it seems to suggest some decision process. Well, no, you can point out --what you've NOT here written, nb-- that you want to preserve joint geomentry. Witness the exclusion by you of one sheet bend EK with U-fold's S.Part qua EK S.Part : unstable.

…

With respect to #1412 Ring bend and its corresponding eye knots... There are 2 [i]linkages [/i]which create 'primary' corresponding eye knots. It turns out that [s]both of[/s] these eye knots are in fact identical. As such, I count that as one (1) - not 2.

Which you then mis-use in subtracting 1 from 8 to assert "7 secondary EKs" when in fact it will be that similar "identical" EKs arise in them to reduce to a trio --half & half.

–dl*

[agent_smith]

Dan,
You seem to focus on syntax/grammar errors - rather than the underlying concepts.
This appears to prevent you from seeing the trees because the forest obscures your view
It appears that you take the extreme literal interpretation of each and every typed word.
The English language is complex - I think you should not jump to a (wrong) conclusion because of a syntax/grammar construction that you perceive to be erroneous.
This makes it very tedious to communicate with you - because I would need to closely examine every possible literal interpretation of typed words - and account for every possible outlier.
In a face-to-face conversion, it would have a retarding effect - because you would need to carefully think about every word you wanted to speak out loud - before uttering those words.
Example: You ask a question… there would be an awkward silence for 15 minutes while I calculate in my mind every conceivable way you could interpret my words, and then I reply.
Then you ask another question, and again you wait in silence for 15 minutes while I calculate another reply… and so on!

For example: (read very carefully)
Below is an example of typing words in the English language to convey a concept.
Q. Can you find different ways to interpret A, B, C, D - and arrive at completely different conclusions?

A. There are 4 segments projecting from any bend.
B. There are 4 projections exiting from any bend.
C. All bends have 2 S.Parts and 2 tail ends.
D. There are 4 segments protruding from a bend - if you ‘cookie cutter’ a bend - you will see the 4 segments.

Another example:

A. There are 4 possible linkages between all segments that protrude from a bend.
B. There are 4 possible linkages that can be made from segments of opposite colour.
C. There are 4 possible linkages that can be made between all of the tails and S.Parts that protrude from a bend.
D. There are 4 possible linkages that can be made between any tail end, and any S.Part.
E. There are 4 possible linkages that can be made between any projection exiting from the core of a bend.
F. There are 4 possible linkages that can be made between any segment of opposite colour that exits from a bend.
and so on…

In my view, there are syntax/grammar differences due to complexities of the English language.
I can see that you appear to adopt an extreme literal interpretation of each and every typed word and sentence.

In terms of A - F above, I can see the underlying concept that is present in each statement.
I imagine that you see conflict in the meaning of each sentence - which is fine - but can you glean any meaning/concepts?
Example:
per Dan Lehman:

you have a linkage between Tails

You have noticed a dotted line in my posted image linking a [b]tail-to-a-tail[/b]. There is also a linkage between an S.Part and an S.Part! Dan - this is intentional. I [i]intended [/i]to link a tail segment to a tail segment! I also intended to link an S.Part to an S.Part! There is no error with this depiction :) It was all intentional :)

With reference to the attached image below:
It is a graphical representation of a ‘bend’.
It shows 4 possible linkages between all of the segments of opposite colour exiting from the core of a ‘bend’.

Possible interpretation risks:
The word ‘linkages’.
The word ‘segments’.
The phrase ‘exiting from the core’.
The phrase ‘opposite colour’.

In one sentence, I have tried to capture the essential concepts - but I appreciate due to the complexities of the English language - different interpretations can be arrived at.
I could add an additional sentence such as; ‘Only segments of opposite colour can be linked’ - to further tighten the definition.

---

[Dan_Lehman]

@agent_smith post:14

per Dan Lehman:

you have a linkage between Tails

You have noticed a dotted line in my posted image linking a [b]tail-to-a-tail[/b]. There is also a linkage between an S.Part and an S.Part! [as I noted} Dan - this is intentional. I [i]intended [/i]to link a tail segment to a tail segment! I also intended to link an S.Part to an S.Part! There is no error with this depiction :) It was all intentional :)

The question wasn’t to your intent, but to your
description of it --to wit (which you conveniently
omitted in quoting me above) ::

My first point: With respect to any bend, there can only be 4 principal 'linkages' [b]between a tail and an S.Part segment[/b] (refer to the graphical representation of the bend below).

Wrong per your image :: you have a linkage between Tails; between "S.Parts & Tail" we might reject the S.P-2-S.P linkage, too --4-2 = 2.

The words "between a tail and an S.Part segment" do NOT include the cases of linkages between Tails & between S.Parts. If you intend to describe the full set of linkages, you should amend this conflicting description to (say) "There are 4 linkages between the Tail & S.Part of one piece to those of the other (not of a piece's own ends to each other)". You might do well to speak here of "ends", noting that ends include Tails & S.Parts. "The are four possible linkages with an end of one piece and that of the other." (This not easy speaking --lotta pitfalls!)

Your C & D are also wrong (D for the same reason as above,
and also denoting like-color’d linkages, as does C).

C. There are 4 possible linkages that can be made between all of the tails and S.Parts that protrude from a bend. D. There are 4 possible linkages that can be made between any tail end, and any S.Part.

Technical writing such as giving descriptions requires adherence
to rules of the language, and imposing clarity where the language
might be ambiguous. And this clarity can be tricky to achieve
(esp. if one acts as one’s own editor).
It bears repeating :: “This not easy speaking --lotta pitfalls!”

–dl*

[agent_smith]

Dan - you reply has confirmed my assertions about your fixation on grammar/syntax - rather than the underlying concept.

We can go back and forth to infinity with these posts.
What do you think this would achieve?

You found this phrase and had a Eureka moment!

between a tail and an S.Part segment

You are intent on drawing attention to grammar - and you cant seem to look past this hurdle. With your pull quotes - what if you just add the letter "s" to tail and S.Part ? - does that change the meaning :) I should have wrote; [b]linkages between protruding segments of opposite colour.[/b] This would have narrowed the number of possible misinterpretations and limited your ability to fixate on peripheral semantics.

Do you think you would ever be able to overcome your fixation on grammar/syntax errors?
Or has permanent and irreversible harm been done ?

Here is what I think.
I think you knew what my intent was - but you like to engage in intellectual debate over semantics.
I don’t mind engaging with you - but it gets very tedious at times.
To be honest with you - I prefer to engage in a way that promotes the advancement of this theoretical subject matter (the relationship between bends and eye knots).
I dislike bogging down on subject matter that is obviously attempting to score goals based on trivial semantics.
Whenever I post info on this IGKT forum - it is within the spirit of advancing the science of knots - not advancing the science of arguing semantics.

Anyhow, it would be nice to see you post your own images of bends and corresponding eye knots!
Have you got any contributions?
Or is this a lost cause?

…

I have uploaded the ‘Reef derived bend’ and corresponding eye knots.
This info graphic does not include all 8 combinations of linkages and loading profiles.
Dont get hung up on this… I dont have time to photograph all conceivable combinations.
I’ve simply included the ‘primary’ corresponding eye knots.
The are 2 'primary corresponding eye knots - and they are identical.

There is a risk of misinterpretation here - due to complexities of language.
eg
There are 2 corresponding primary eye knots - but they are identical.
There is an opportunity to debate this… do you count it as ‘1’ or ‘2’?
If I say ‘1’ this opens up a can of worms.
If I say ‘2’ - maybe less worms in the can
To reduce language semantic arguments, I’ll just say:
“There are 2 primary corresponding eye knots - both of which are identical”.
This should limit the number of possible interpretations?

---

[Dan_Lehman]

@agent_smith post:16

We can go back and forth to infinity with these posts.
What do you think this would achieve?

What is hoped to be achieved is corrected/improved
descriptive material, presentation of ideas.

You are intent on drawing attention to grammar - and you cant seem to look past this hurdle. /// Do you think you would ever be able to overcome your fixation on grammar/syntax errors? /// I dislike bogging down on subject matter that is obviously [b]attempting to score goals[/b] based on trivial semantics.

THIS is your fixation --that discussion is about "scoring goals". And your defensive reaction comes as irony to your own replies taking others to task for, say --one of your favorites-- writing "THE bowline" (emphasis yours), instead of explicitly & exactly stating, in your desired language, "#1010" !

[u]Here is what I think.[/u] I think you knew what my intent was - but you like to engage in intellectual debate over semantics.

Why didn't you instead think "He's found a point where I've not clearly expressed my ideas --good, I'll fix that." nb : "fix THAT" --not "fix HIM"!

And I have been contributing some good ideas in this discussion,
but you assiduously avoid them.

Every simple --i.e., of two pieces-- ends joint defines
a 2-tangle, in which one piece, with ends “1” & “2”,
entangles 2nd piece, with ends “A” & “B” such that
“1” is loaded in opposition to “A” --the end joint’s S.Parts–
and ends “B” & “2” are its Tails.

A 2-tangle has 8 loading profiles that potentially
define eye knots; these are (with reference to the
ends joint’s S.Parts & Tails) :

----------S.Part-loaded [parens of ends joined into eye ]

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

These loadings “potentially” create viable eye knots;
but they might lack stability to sustain such a loading.
Also, with symmetric structures, there will be duplicate
loadings yielding a given knot (e.g., the 1st & 3rd and
2nd & 4th loadings pairs will each yield one knot, not two).

… and so on.

I’m still finding trouble getting around variations in the
geometry, in the dressing (emphasis, perhaps : “heavy
dressing” !) of knots to fit into, to flow from Tangles.
E.g., what is the (<–presumptive singularity?!) net knot
from the capsized Carrick bend tangle (which is some
remove from the lattice-form) --one can take the joint’s
Tails in various directions. (To the point re singular vs.
plural “net knot” it’s a question now of angles of loading,
not merely of ends.)

–dl*

[agent_smith]

Continuing with the development of bends and their corresponding eye knots…

I’ve uploaded the Lehman8 variation bend
Before any nuclear melt downs occur - note the typed word ‘variation’.
Its not the original Lehman8 - its a variant.

I’m examining some loading profile characteristics between these 2 ‘bends’ and their corresponding eye knots.
Interestingly, both of these bends can produce ‘offset’ Lehman8 bends.
I’m currently comparing the stability, security, and jam resistance of these 2 ‘offset’ bends.

And thank you Dan for a civilised reply - there seems to be hope that 2 knot geeks can find common ground
I note your alpha numeric system to denote possible combinations of Tail-S.Part-S.Part-Tail linkages + loading profiles.
I understand it - because I already understand the underlying concept.
However, the casual/average reader might scratch his head for a while trying to decipher the meaning.
That’s why I prefer a graphic representation of a ‘bend’ (refer to my attached image below).
The colour coded image makes it clear and easy to understand that there are 4 combinations of possible linkages.
(And that linkages can only occur between segments of opposite colour).

As far as the current state of progress on this theoretical work:
I seem to be settling on a few key definitions - which I can now advance herein:

1. For any given ‘bend’, there are a 4 possible combinations of linkages between the segments protruding from the knot core (refer to colour coded image below).
2. These 4 linkages produce corresponding eye knots.
3. For each corresponding eye knot, there is a choice of 2 S.Part’s - which therefore produce a total of 8 possible corresponding eye knots (based on loading profiles).
4. Of the 8 possible corresponding eye knots, up to a maximum of 2 will be ‘primary’.
5. The ‘primary’ corresponding eye knots are closest to the parent bend in terms of loading profile - loaded segments of the eye knot are congruent with the parent bend S.Parts (refer to coloured dots in attached image).

…

With regard to your comment re “THE Bowline”
I can respond as follows…

There are many different type of ‘Bowlines’.
If a person types ‘The Bowline’ - it immediately begs the question… “Which Bowline are you referring to?”
Many climbers are guilty of this - because some have an underlying agenda to attack and denigrate the use of ‘the’ Bowline as a tie-in knot to a harness.
In fact, several types of ‘Bowlines’ are inherently secure (and some aren’t).
We know that #1010 Simple Bowline in not ‘safe’ as a tie-in to a climbers harness.
But that is not the principal purpose of #1010 (ie its not intended for life critical applications).

Per your other points:

These loadings "potentially" create viable eye knots; but they might lack stability to sustain such a loading.

I've already pointed this out several times... not all of the 8 possible corresponding eye knots will be stable (or 'viable' as you have typed). In most cases, I think it is possible to derive up to 4 stable corresponding eye knots - but this is not a hard and fast rule. I am of the view that it all comes down to the core structure of the [i]parent bend[/i]. For example, the Reef/Square bend does not produce any stable corresponding eye knots. This due to the union of 2 'U folds' (or 2 bights). A phenomena called 'bight straightening' occurs.

The Sheet bend does produce 4 stable (viable) corresponding eye knots, but it appears that the remaining 4 are unstable (due to bight straightening).
Note: This is because the ‘Sheet bend’ is the union of a ‘U fold’ + a loop.

The Carrick bend is the union of 2 Crossing hitches - and this produces some stable/viable corresponding eye knots.
However, I am of the view that some of the 8 corresponding eye knots may be unstable (again due to risk of bight straightening).

And this last remark:

I'm still finding trouble getting around variations in the geometry, in the dressing (emphasis, perhaps : "heavy dressing" !) of knots to fit into, to flow from Tangles. E.g., what is the (<--presumptive singularity?!) net knot from the capsized Carrick bend tangle (which is some remove from the lattice-form) --one can take the joint's Tails in various directions. (To the point re singular vs. plural "net knot" it's a question now of angles of loading, not merely of ends.)

In my view, the simplest way to look at this is from a geometric perspective. The corresponding eye knots must have some congruence with the parent bend geometry. The cores of all 8 corresponding eye knots must match the geometry of the [u]parent bend core[/u]. Obviously, under heavy load, some of the corresponding eye knots will deform/distort - because they are not stable in a particular loading profile. I don't see this as a show-stopper - because its an artefact of loading.

And note:
Force is injected into the core of a bend from 2 opposing directions.
Force is injected into the core of an eye knot from 3 directions (2 eye legs in opposition to 1 S.Part).
Loading profile matters…

---

[Dan_Lehman]

@agent_smith post:18

Continuing with the development of bends and their corresponding eye knots…

Here I’ll again interject that correspondence can --even should?!–
have different meanings. I.p., I pointed to the process of building
an EK from a e2e Joint by using 1 single end joining to the two
(“twin”) ends of a U-fold, and then fusing the single end into one of
the others so to make a 2-tangle (unfused :: 3-tangle) and EK.

I think that we’re best suited to speak in terms of pure logic of
the ends of 2-tangles (my main focus : amply but not too involved)
vs. more value-connoting terms such as “primary/secondary”.
Perhaps these values will accrue to one or another knot,
got via one or another logical relationship. And such wording
be more sensible with knots-users than Ivory Tower logics!?

The SheetBend/BWL/… structure is beguiling to the point
I’ve mused about “specious” --both simple and accommodating
of a planar graphical image, and nicely asymmetric. To my
thinking via the construction of an EK given above, Ashley’s
#1043/5 & #1016 are all corresponding knots to the Sheet
Bend --in one way more so than the BWL :: in that these
knots do NOT have Xarax’s pure “nipping loop” : they
have a loop loaded principally on the S.Part end, but
don’t run then out into a half-force-loaded Outgoing Eye Leg!

I note your alpha numeric system to denote possible combinations of Tail-S.Part-S.Part-Tail linkages + loading profiles. I understand it - because I already understand the underlying concept. However, the casual/average reader might scratch his head for a while trying to decipher the meaning.

One more reason to wear a helmet! The 2-tangle shows the 4 ends, and they get labelled somehow --simpler to do with an EK since it has by one S.Part (if one has feelings of primary/secondary connotations resp. of "1/2"! :-). Doing with them comes via Loading Profile.

Which all seems neat & clean/perspicuous, befitting logic;
then the struggle to fit reality of physics --dressing & setting–
becomes an issue. yChan’s particular dressing of #1425a
is one eye opener; the Tail-Loaded BWL coming out of what
I’d thought was so accommodating of the BWL 2-tangle is
another provocative case.

The colour coded image makes it clear and easy to understand that there are 4 combinations of possible [b][i]linkages[/i][/b]. (And that [u][i]linkages [/i]can only occur between segments of opposite colour[/u]).

Which puts the like-connected, same-colored cases of what one might call the "Becket Hitch" in some difference to the actual structure --which might not [u]connect[/u] the two same-colored ends (for that matter, my 2-tangle EKs don't have "connected" ends --just equally loaded opposing the S.Part(!)--, though here there can be real-knot differences (loading being NOT equal, though opposing : e.g., 100% S.Part vs. 35%+65% legs).

[u]And note:[/u] [ ] Force is injected into the core of a bend from 2 opposing directions. [ ] Force is injected into the core of an eye knot from 3 directions (2 eye legs in opposition to 1 S.Part). Loading profile matters...

Imagine an OWK ("EDK", "Offset Water Knot") joining two ropes that are hanging from a narrow strip of steel:: and perhaps someone abseils or ascends these ropes. What type of knot is that? (It doesn't have force from opposite directions but the same, on either side of the steel strip.

While we’re into this fun&games corner of knotting,
what type of knot is the OH structure commonly
tied in one’s plastic film produce bag, to keep the,
um, broccoli from spilling out?! --or esp. when the
bag (with contents or just air!) is compressed as to
pop it?!

–dl*

[agent_smith]

Dan,

To my thinking via the construction of an EK given above, Ashley's #1043/5 & #1016 are all corresponding knots to the Sheet Bend --in one way more so than the BWL :: in that these knots do NOT have Xarax's *pure* "nipping loop" : they have a loop loaded principally on the S.Part end, but don't run then out into a half-force-loaded Outgoing Eye Leg!

I've attached an image of the #1043 [i]F8-Loop bend[/i] and corresponding eye knots. I don't see how #1043 knot core is the same as #1431 Sheet bend core - unless I've misunderstood you? The notional concept of 'correspondence' is only valid from a strictly geometric point-of-view; where the cores of the eye knots are geometrically identical to the core of the parent bend. If you change the geometry, you change the correspondence to something else.

#1016 if essentially the same as #1074 (with the only difference being the positioning of the ‘bight-tail’ end.
If we build an end-to-end joint (bend) with a core that geometrically matches the core of #1016/#1074, it isn’t the same as a Sheet bend.
I don’t have a photo… but if I find time I will try to upload something.

Having said that…
#1074 Bowline with-a-bight (and #1016) and #1080 Bowline on a bight are all eye knots with interesting structures.
I agree that they don’t fit the non TIB tied end-of-line geometric form where there are 3 (loaded) segments exiting from the knot core.
Okay - #1080 Bowline on a bight actually has 6 segments exiting from the core (but only 5 segments are loadable when there is a designated tail end protruding).
#1074 also has 6 segments (and so does #1016).- but only 5 of these segments are loaded/loadable when there is a tail end present).
What is the common denominator with these structures?
Well, they are all “TIB” (Tiable In the Bight).
So:
A ‘non TIB’ eye knot tied end of line has 4 segments, 3 of which are loaded/loadable when there is a designated tail end.
A ‘TIB’ eye knot often has 6 segments 5 of which are loaded/loadable when there is a distinct tail end.

Although;
#1043 eye knot is ‘TIB’ and has a total of 4 segments exiting the core (3 of which are loaded or loadable)
#1058 directional F8 is also ‘TIB’ and it only has 4 segments exiting the core (but only 3 of which are loaded/loadable).

What does this all mean?
I don’t have all the answers!

Here I'll again interject that correspondence can --even should?!-- have different meanings.

Not for me - sorry. I am of the view that terminology and concepts need to be defined - so as to remove as much ambiguity as reasonably possible. Words have meaning - otherwise it would be like the tower of Babylon and people would not be able to communicate.

I pointed to the process of building an EK from a e2e Joint by using 1 single end joining to the two ("twin") ends of a U-fold, and then fusing the single end into one of the others so to make a 2-tangle (unfused :: 3-tangle) and EK.

You like thought experiments that employ 'edge cases'. I like thought experiments - but they have to be grounded in reality. In my view, an end-to-end join will have 4 segments projecting from the knot core (and an 'eye knot' also has 4 segments). There is no 'fusing' in a real-world practical case where a person joins 2 ropes together by hand alone. If this was done (by splicing or by swaging) - in my view it falls outside the realm of ordinary knotting. Same with 'beckets' and 'splices' - it is outside the scope of my work.

OR;
You possibly mean tying a bend using 2 ‘bights’ (2 U folds).
eg;
If I create a Reef/Square knot with 2 bights, this means 8 segments exiting from the core (an even number again).
If a Sheet bend is tied with 2 bights, this has 6 segments exiting the core (an even number).
Its an interesting thought experiment

I probably need to tighten the language construction to confine a knot structure to within specific parameters.
Eg;
A ‘bend’ that is formed from 2 singular rope ends - no bights.
This would have 4 rope segments exiting from the core (an even number).
A non-TIB ‘eye knot’ that is tied end-of-line from a singular rope (this is a bit wordy).
This would only have 4 rope segments exiting from the core - 3 of which are loaded (an odd number).

If think the overall message here is that eye knots and bends can be tied-with-bights (and this is your main point?).

I think a generic distinction is that a bend has 2 loaded segments (‘S.Parts’) acting in opposition - when there are 2 distinct tail ends.
An eye knot has 3 loaded segments - when there is a distinct tail end - the 2 legs of the ‘eye’ are in opposition to the S.Part.
But all of this is in respect of loading profile…

Imagine an OWK ("EDK", "Offset Water Knot") joining two ropes that are hanging from a narrow strip of steel:: and perhaps someone abseils or ascends these ropes. What type of knot is that? (It doesn't have force from opposite directions but the same, on either side of the steel strip.

I don't follow this... it seems to be another 'edge case'? I'm trying to visualise 2 ropes that are hanging from a narrow strip of steel (likely not load rated?). And these 2 ropes are then united with #1010 Offset overhand bend? And then a person abseils/ascends on these joined ropes? [i]No comprende[/i]...sorry.

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