The relationship between 'bends' and eye knots

[Dan_Lehman]

With regard to the 'correspondence between bends and eye knots':

Clearly - I am biased because I do believe there is a relationship between a 'parent bend' and its offspring 'eye knots'.

I will simply again point out that w/o throwing anything
out of consideration (with a focused view) one has, from
whatever *knot* is up for consideration, a *tangle*;
and a *tangle* has all these possibilities re its *knots*
via the loading profiles.

But formalizing this is really tough.  That yChan
dressing of SmitHunter's Bend quite impressed me
--the "same" knot?!  Not by my reckoning, but then
it shows how differences can sneak around formal
barriers.

[HINT]: #1425 is topologically equivalent to the 'False Zeppelin bend' (with crossed tail segments).

And I'd like to see its performance for strength.
We've seen Thrun's Joint/PoorMan'sPride/zep. go out
of form, in the HowNotTo guy's video (amazing!).

--dl*
====

[agent_smith]

from Dan:

I will simply again point out that w/o throwing anything
out of consideration (with a focused view) one has, from
whatever *knot* is up for consideration, a *tangle*;
and a *tangle* has all these possibilities re its *knots*
via the loading profiles.

Am unclear as to what the ultimate purpose of this comment is?
I am not being offensive - I am simply being factual - in that I am left wondering what your point is?
Ok.. the 'knot' that is up for consideration is presumably a 'bend' (an end-to-end join).
I'll refer to this "knot" (a tangle per your words) as the 'parent bend'.
Note: Definition of 'tangle' = a confused mass of something twisted together
I am of the view that any knot that is tied with an intended specific geometry is not a confused mass.

per Dan:

But formalizing this is really tough.

I am making the following claim:
A deliberately tied parent bend which has a specific intentional geometry has a defined number of [derived] corresponding eye knots.
These corresponding eye knots have a [core] geometry that is congruent with the [core] geometry of the parent bend.
Note: I am using the word 'core' to denote the part of the knot that is central to its existence or character.
I find this definition better than 'nub'.

I am using the term 'eye knot' in lieu of loop knot.
An eye knot is analogous to an eye bolt - the eye being round/oval and permits connectivity (eg a carabiner).
The 'eye' of an 'eye knot' has no particular chirality (handedness).

I am (by definition) making the claim that the correspondence (or relationship) between a parent bend and its derived eye knots is geometric in character.
This claim is valid where load (or a loading profile) is not considered.
That is, the correspondence is purely geometric in character.
All knots respond to load in different ways - and the particular loading profile plays a significant role in this 'response'.
Typical responses include (list is not exhaustive):
[ ] compression
[ ] distortion
[ ] extrusion of rope segments out of the core
[ ] instability
[ ] insecurity
[ ] collapse

Dan Lehman may wish to make an alternative claim (or perhaps make no claim whatsoever).
He is entitled to do so.
Dan may wish to reject a geometric relationship and consider loading profile as the dominant factor to consider.
If this is his preferred approach - this may explain his reference to "possibilities".
There may indeed be a number of possible 'eye knot' derivatives - perhaps being difficult to quantify?
In contrast, if a geometric approach is taken, it ought to be easier to quantify the number of possible eye knot derivatives.
And again - my preferred approach is to begin with [a] 'bend' and then try to derive the corresponding eye knots (rather than the other way around).
In my view, the logical approach is to use a 'parent bend' as the basis for deriving corresponding 'eye knots'.

That yChan
dressing of SmitHunter's Bend quite impressed me
--the "same" knot?!  Not by my reckoning, but then
it shows how differences can sneak around formal
barriers.

In my view, this is potentially straying off-topic - and is best examined in a separate topic thread.

Quote

"[HINT]: #1425 is topologically equivalent to the 'False Zeppelin bend' (with crossed tail segments)."

And I'd like to see its performance for strength.
We've seen Thrun's Joint/PoorMan'sPride/zep. go out
of form, in the HowNotTo guy's video (amazing!).

Possibly best dealt with in a new topic thread.
But, 'strength' is irrelevant in my view.
Possibly you meant some kind of 'response to load'?
Response to load (or from some particular loading profile) allows one to assess things like; stability, security, jam resistance, etc.
For example, strength is not a relevant factor in the use of #1410 Offset overhand bend in abseiling/rappelling.
Focussing on 'strength' leads one to wrong conclusions (because stability, security, knot footprint, and jam resistance are more important factors).

[Dan_Lehman]

from Dan:

I will simply again point out that w/o throwing anything
out of consideration (with a focused view) one has, from
whatever *knot* is up for consideration, a *tangle*;
and a *tangle* has all these possibilities re its *knots*
via the loading profiles.

Am unclear as to what the ultimate purpose of this comment is?

I mean to push towards reaching the Tangle level
and then going from there with whatever particular
loading profiles (eye knots, e2e joints, knot hitches...)
are of interest.  Whereas, in this ...

I find it logical to begin with a 'bend'
- and then try to derive the corresponding eye knots.

... I felt a bit constrained.  In a sense, one isn't
so much deriving ... but showing --they're
right there, in the Tangle, to be ID'd per the Tangle
pieces (1-2 & A-B, for a 2-Tangle).

Note: Definition of 'tangle' = a confused mass of something twisted together
I am of the view that any knot that is tied with an intended specific geometry is not a confused mass.

... and that of course we're not using "Tangle"to mean this,
but simply an entanglement of cordage devoid of loading.
(Deliberateness might be overrated; in any case, one has
THIS or that Tangle, however wrought, for the consideration
of its various Knots via loading.

Note: I am using the word 'core' to denote the part of the knot
that is central to its existence or character.
I find this definition better than 'nub'.

What happens if "core/nub" is omitted --esp. re e2e joints!?

That yChan
dressing of SmitHunter's Bend quite impressed me
--the "same" knot?!  Not by my reckoning, but then
it shows how differences can sneak around formal
barriers.

In my view, this is potentially straying off-topic - and is best examined in a separate topic thread.

Well, it points out a significant change of your
vaunted geometry in a knot that is put together
able to have such difference.
(Consider also how there are different dressings
of the Fig.9 EK.)


--dl*
====

[agent_smith]

This post is directed to Dan:

We are getting technical here...but it is useful to sort out definitions... because I think this is fundamental to any progress.
Also, this is a forum for technical discussion about knots - with deep dives into underlying concepts.
I don't know of any other forum on planet Earth where these types of peer-level discussions are possible?
In this regard, I'm happy to explore this subject matter with a view to finding some common understanding and agreeance.
At the very heart of this subject matter is the English words we choose to convey complex ideas...Words have meaning.

In reply to your points...

I mean to push towards reaching the Tangle level
and then going from there with whatever particular
loading profiles

We need to agree on definitions.
I hold the view that the word 'tangle' has a particular meaning as follows:
"a confused mass of something twisted together".
A random knot that accidentally formed in a rope might fit this definition (eg due to wind, strong hydraulic water flow in a stream, shaking/jostling, etc).
However, I am strongly of the opinion that when a person ties a knot he has a particular outcome and purpose in mind.
The knot tier directs his mind to the task - applying cognition - to achieve a particular and/or specific geometric outcome.
For example, I intend to tie a Butterfly eye knot (Ashley #1053).
The result I end up with can be verified against a known standard (I can look up #1053 in a book and verify it).
The end result (ie outcome) is either correct or incorrect. I am of the opinion that you can't say that a knot is half correct (or 25% correct).

There must be some basis for establishing and agreeing upon certain standard geometric knot forms - for example, we all agree that a simple Bowline is depicted at #1010 in Ashley's book.
In the same way, we have the 'SI' metric units of measurement - and we all agree on what a metre length is. And we all agree on what 1 kilogram mass is.
Without a standard model to compare against, there would be no way for an assessor to assess a trainee. A trainee could tie anything into a confused mass of twisted rope and demand to pass an assessment.
For example: An assessor asks a trainee to tie #1053 Butterfly. The trainee presents a confused twisted mass of rope to his assessor.
The trainee demands to be assessed as competent.
An assessor needs measurable criteria to enable decisions/judgements to be made about a student competence.
However, for this to be true - there must be an agreed standard for a knot that is assigned the name 'Butterfly'.
Without an agreed standard - there can be no agreement - only confusion.
EDIT NOTE:
Can we assume Ashley and CL Day to be primary reference sources?
If I look up a knot that is assigned the name 'Bowline' - I anticipate finding the exact likeness as depicted at illustration #1010 in ABoK.
I would not expect to find the knot illustrated at #1047 (F8).
Humans assign names to things - eg I know what a tree looks like, and a chair, and a dog, etc.
If you asked me to tie a Zeppelin bend, I would assume that you had a specific geometry in mind - it wouldn't be in the likeness of #1415 Double Fishermans?
If I tied and presented #1415 (instead of a Zeppelin bend) - I would surmise that you would look at me in astonishment?

You yourself have complained about which geometry is the 'ASCii Bowline'.
You sent me a number of emails with ASCii code in an attempt to depict a certain knot geometry.
That is, you had a particular geometry in mind.

... I felt a bit constrained.  In a sense, one isn't
so much deriving ... but showing --they're
right there, in the Tangle, to be ID'd per the Tangle
pieces(1-2 & A-B, for a 2-Tangle)

In order to show or demonstrate that something is being related to something else - one must derive it from a 'source'.
For me, the 'source' is the 'parent bend'.
I begin with a parent bend - and I derive the corresponding eye knots by linking the Tail(s) and S.Part(s) in various combinations.
I had posited that there are only 4 possible linkages that can be made.

Per your words... "showing they're right there" suggests that you were able to identify something visually.
You observed some kind of congruence or correspondence from one thing compared to the other.
Could this "showing" be geometric in character? How else would you be able to "show' something?
I had posited that it is the core structures that can be compared - the correspondence being that the cores have the same geometry.
Here again I use the word 'core' as my preferred way of identifying the 'nucleus' of a knot structure.
I define a knot core to denote the part of the knot that is central to its existence or character.

...per the Tangle pieces (1-2 & A-B, for a 2-Tangle)

I think you mean the segments that protrude/project from the knot core?
The 1-2 and A-B don't have a clear-cut definition - hard for a layperson to understand (you would need to establish definitions).
Presumably you refer to the 4 segments (protuberances) exiting the knot core - and you are assigning an alpha-numeric coding to ID these segments?
In 3D space, which orientation gives rise to assigning a particular segment as a "1" in contrast to a "2", or "A/B"?
In other words, how do I determine which protruding segment is "1"?
EDIT:
Given that a 'bend' has 2 Tails and 2 S.Parts...
Perhaps assigning the following alpha-numeric values might make it easier to understand:
S1, S2 and T1, T2
S1 = "S.Part 1" (one of the standing parts)
S2 = "S.Part 2" (the opposite standing part)
T1 = "Tail 1" (one of the tails)
T2 = "Tail 2" (the opposite tail)

S1 can link to S2 (S1-S2)
T1 can link to T2 (T1-T2)
S1 can link to T2 (S1-T2)
S2 can link to T1 (S2-T1)
*Note that S1 cannot link to T1 (both originate from the same rope)
*And S2 cannot link to T2 (both originate from the same rope).

I find this alpha-numeric system and my descriptions to be more 'logical' than your 1-2 / A-B annotation.
You may choose to disagree (and that's fine).

What happens if "core/nub" is omitted --esp. re e2e joints!?

In a 'bend', there will be 2 S.Parts and 2 Tails?
I think you might also have contemplated another type of 'bend' which is an edge case - eg the linking of 2 eye knots?
For example; I could link the eye of a #1010 simple Bowline to the eye of another #1010 simple Bowline.
Would this be your alternative definition of a 'bend'?
For me, I see this as a composite union of 2 eye knots - with each eye knot possessing its own core.
Perhaps a stricter definition of a 'bend' is that the union only has 1 core (not 2 separate cores).
In the case of 2 cores (linking 2 eye knots) - this is a composite structure - and each knot core will respond to load accordingly.
[it could be the linkage of #1010 simple Bowline to #1047 F8...an eye-to-eye link... Each eye knot responds differently to load.]

Well, it points out a significant change of your
vaunted geometry in a knot that is put together
able to have such difference.
(Consider also how there are different dressings
of the Fig.9 EK.)

I note the use of "your vaunted geometry" phrase.
It isn't 'vaunted' per se - its simply a logical choice.
If I tie a Zeppelin bend - it will have a known geometry.
(although again, there must first be an agreed definition/geometry for what constitutes a Zeppelin bend).
If I tie #1411 F8 bend - it has a known geometry.
Here I assume the energy stable dressings - the simplest most symmetric dressing that is demonstrably stable under load.
With an 'F9', its simply a matter of declaring a particular dressing - again - the most energy stable dressing is logical.
For all 'bends' and 'eye knots' one must settle on a dressing - nominally the most energy stable dressing.
For example: An F8 bend (#1411) can be tied with a flat parallel dressing state - but this is unstable.
One can also apply the same general principle to an offset joining knot - eg #1410 - where some dressings will be more unstable.

Again - we need to have agreed standard to reference against.
An F8 and an F9 eye knot can have different dressing states.
However, we know that there are geometries that are more stable in response to load.
Example:
A trainee is asked to tie an F8 eye knot.
The trainee presents the F8 to his assessor.
What is the criteria the assessor is making judgements about?
What evidence does an assessor require to form a judgment about the trainees competence?
What evidence is required for an assessor to declare a trainee 'competent'?
Note: Assessment should also capture consistency of performance - to rule out random chance success, the trainee should accurately tie the F8 at least 3 times.
Accuracy and consistency ought to be part of the assessment criteria.

EDIT NOTE:
I've added an image to illustrate the concept of a standard reference.
An assessor could look up what an 'F8 bend' looks like in a primary source (eg Ashley).
An F8 bend is found at illustration number One thousand four hundred and eleven (#1411).
However, Ashley does not define what an energy stable dressing is... one can only assume that the depicted dressing is 'optimal' for loading.
I define an energy stable dressing state as:
"Being optimal for loading - to the extent that the knot remains stable and exhibits the least degree of distortion in response to load".
With respect to the attached image below: Image 'B' is energy stable.

[Dennis Pence]

I know this is an old topic, but I just wanted to add one more example of what I think agent_smith is trying to show.  Here is the basic structure that can be found in the Hanson patent document for what I like to call the Hanson Knot Family.  {I have deleted the labels which were the 12 points of the BSA Scout Law for simplicity, but the Figure numbers come from the patent.}

[Dennis Pence]

Here are the 8 different loop knots that can be derived from the original Hanson Structure.  Alden Hanson had 3 of them in his patent application, and I think that these are the best three from the eight.

The "sideways" loops in the last four become almost unrecognizable when you tighten them, and I am not sure they are very practical.

[agent_smith]

Hello and thanks Dennis.

With regard to your mentioning of the number "8", I can comment as follows:

A parent end-to-end joining knot (ie 'bend') has 4 corresponding principal eye knots that are derived by the linkages between available tail ends and S.Parts.
This is true for all end-to-end joints where there are 2 S.Parts and 2 tail ends.

With each of these 4 principal eye knots, the S.Parts can be transposed with the tails (a transposition).
This transposition results in a further 4 eye knots.

If we consider the 4 principal corresponding eye knots (created from the 4 linkages), + the 4 transposed eye knots, this means a total of 8 possible corresponding eye knots derived from a singular 'parent bend'.

NOTES:
1. I prefer to start with an end-to-end joint (the 'parent bend'), and then derive its 4 principal corresponding eye knots.
2.  It is also possible to begin with an eye knot, and then derive [a] corresponding 'bend' and its other related eye knots.
    I personally find this approach less intuitive with multiple possible pathways (although Dan Lehman appears to favour this approach?).
3. The correspondence is geometric in nature (the core of the 'bend' is congruent with the cores of its derived eye knots).
4. I am using the term 'eye knot' in lieu of 'loop knot' (a 'loop' has chirality - S/Z - an 'eye' has no chirality).
5. There is no peer reviewed primary reference source for the relationship between 'bends' and 'eye knots' (the concept of correspondence has been known but not documented in detail).
6. Harry Asher may have been first to try to publish a basic theory of correspondence between 'bends' and 'eye knots' - Ashley apparently did not?

[Dennis Pence]

For a more definite reference, I have a copy of Harry Asher's The Alternative Knot Book (1989), where there is one section on page 81 titled "Loops from Bends."  His primary example is the Corrick Bend (which he claims is unnamed in Ashley, but I could not quickly find it).  He gives only one "loop", but given the extreme symmetry, there would not be 8 different eye knots that could be derived (but certainly more than one).  On the next page, he mentions several other bends that may or may not lend themselves to this process.  Also on page 82, Asher has a section titled "Bends from Loops."  Here he writes "The reverse procedure is vastly simpler."  His only example is the Angler's Loop, and he suggests two different cuts to derive two different bends from this loop.

I have also looked at the different bends that you can derive from the Hanson Knot Structure.  If you start with the basic Handson Bend (which Heinz Prohaska called a Ram Head Knot), you can switch the standing part and the free end on one rope (but not the other) to get one offset bend.  Then you can switch the second rope (but not the first) to get a second offset bend.  Finally, you can switch the standing part and the free end of both ropes to get the fourth different bend.  Hanson did not have all of these clearly drawn in his patent document, but he seemed to imply them all.

[Dan_Lehman]

With regard to your mentioning of the number "8", I can comment as follows:

A parent end-to-end joining knot (i.e. 'bend') has
4 corresponding principal eye knots that are derived by
the linkages between available tail ends and S.Parts.
This is true for all end-to-end joints where there are 2 S.Parts and 2 tail ends.

But it's not true in terms of distinct *knots* if the Tangle
is symmetric --for you'll just get duplicates (which I will
count as such even were they to have different handedness).

With each of these 4 principal eye knots, the S.Parts can
be transposed with the tails (a transposition).
This transposition results in a further 4 eye knots.

Which for me isn't transposition but a change in Loading
Profile applied to the 1-2 & A-B (the two pieces) 2-Tangle.

NOTES:
1. I prefer to start with an end-to-end joint (the 'parent bend'), and then derive its 4 principal corresponding eye knots.
2.  It is also possible to begin with an eye knot, and then derive [a] corresponding 'bend' and its other related eye knots.
    I personally find this approach less intuitive with multiple possible pathways (although Dan Lehman appears to favour this approach?).

Wellll, one nice thing is to have a single S.Part --a key part!
But things are really not so simple, doable.  E.g., for our good
ol' friend the (basic) BWL #1010, with S.Part "1", Outgoing Eye Leg "2",
Returning Eye Leg "A" & Tail "B",
how does one apply the Loading Profile of B-vs-2+A (i.e., this  [<<corrected "B-vs-2+A]
is a "Tail-Loaded BWL") ?!!  Taking the Tail one way will yield
an amphichiral knot, and geometries of the B-A part can differ!
<sigh>

--dl*
====

[agent_smith]

Hello Dan,

I had posited that 4 principal eye knots can be derived from a 'Bend' (they correspond to the parent bend).
This is due to the available 'linkages' that can be made between S.Parts and Tails.

With regard to your comment:

But it's not true in terms of distinct *knots* if the Tangle
is symmetric --for you'll just get duplicates (which I will
count as such even were they to have different handedness).

I disagree.
Lets examine Ashley's #1411 F8 bend and its corresponding 4 eye knots.
I would describe this bend as being 'symmetrical'.
The 4 corresponding eye knots are 'distinct'.
And here we need to precisely define what you mean by distinct.
You did comment that you would "count as such even were they to have different handedness".
Which appears to conflict with your first statement.
English language is complex - it is possible to arrive at 2 different conclusions with
what you wrote?

I had already posted details of the eye knots that correspond to #1411 F8 bend.
It is true that 2 of the eye knots have opposite chirality (handedness).
Question: If 2 F8 eye knots have opposite chirality, are they 'distinct' from each other?
(that is, can the F8 eye knots be regarded as geometrically different?).
Or - are they the same?
That is, can an argument be made that they are identical?
I posit that no, they are not identical.
They are mirrors of each other - one is the opposite reflection of the other.
They are both of the same species (F8 eye knots) but they have differences in orientation.
I see this with humans - we are all from the human race.
But, all humans have differing appearance (unless they are identical twins).

The same applies to the Zeppelin bend and its 4 corresponding eye knots.
Two (2) of these eye knots have opposite chirality.

#1425A Riggers bend is a curiosity.
It of course has 4 corresponding eye knots.
BUT, two (2) of these eye knots are identical.
They have the same geometry - and same chirality.
Perfect identical twins.

The question for me is Why?
Why does the Riggers bend have this property?
Why are the Zeppelin bend's corresponding eye knots all 'different' (ie no identical twins)?

#1415 Double Fishermans bend is another curiosity.
It has four (4) corresponding eye knots.
BUT, it has 2 sets of twins!
That is, 2 of the corresponding eye knots are identical twins.
AND, the other 2 corresponding eye knots are also identical twins.
From my research so far, #1415 Double Fishermans bend is the only knot that produces
a double set of identical twins.

I have no theory to explain this.
eg Why does the Riggers bend have a pair of identical twins, while the
Zeppelin bend does not?
Why does the Double Fishermans bend have 2 sets of identical twins?

...

I am still trying to figure all of this out...
#1415 Double Fishermans bend is symmetric (but its geometric shape differs from one side relative to the other).
The F8 bend is also symmetric, but one side is not identical to the other.
The chirality is unchanged, but the positions of the 'collars' relative the S.Parts has changed.

Hmmm.

[Dan_Lehman]

#1415 Double Fishermans bend is another curiosity.
It has four (4) corresponding eye knots.
BUT, it has 2 sets of twins!
That is, 2 of the corresponding eye knots are identical twins.
AND, the other 2 corresponding eye knots are also identical twins.
From my research so far, #1415 Double Fishermans bend is the only knot that produces
a double set of identical twins.

Whoa :: since your making of eye knots only from
connecting (variously) ends of the joint --and not
altering the *tangle*/nub--, how can the Grapevine
joint be any different in what you're claiming than
the Fisherman's joint, or Triple Fish/Dbl.G. ?!
(Try showing images for this!)

--dl*
====

[agent_smith]

Whoa there Dan!
Not sure what you're Whoa'ing about?

With respect to Double Fishermans bend (Ashley #1415) - it is a curiosity for me.
As with all end-to-end joining knots (ie 'bends'), it is possible to derive 4 corresponding eye knots.
A curiosity with Ashley #1415 is that it produces 2 sets of identical twins 'eye knots'.

Have you examined this yourself?
Or do you just like to 'Whoa'?

If I find time today, I'll post some photos.
Presumably, you need photographic proof?

EDIT NOTE:
I also reiterate again:
With respect to the Zeppelin bend and Riggers bend (Ashley #1425A):
[ ] Riggers bend produces a set of identical twin eye knots
[ ] Zeppelin bend does not

I stated previously that I don't know why.
As of today Oct 01 2024, I still dont have a solid theory to explain why.