The Zeppelin Knot is identifiable as a fail and degenerate Carrick Mat

[Stagehand]

The Zeppelin Knot is identifiable as a fail and degenerate Carrick Mat.  When Carrick Mat (Rolfsen 8_18) is used correctly as a bend or as a fixed loop, its tying pattern has a different outline than as presented in the I.G.K.T. logo, the Flat Lanyard Knot, or the Same-side Carrick Bend.  The lines of force as used by the square antiprism knot, when used as a bend or a fixed loop, have the presentation of a three-lobed mat rather than the more typical four-lobe presentation.
Without confusion, this 'corrected' three-lobe pattern of the Carrick Mat has similarities to the typical Carrick pattern or Josephine which is best represented in the Carrick Bend.  A feature of the Josephine pattern is that it is so readily found as a natural knot form that many similar knots are given the Carrick name. Yet, excepting a True Carrick Bend, these knots are all collectively fail for each lacking complete over-under patterning and are all collectively degenerate for each having at least one twist between two lines.
Moving away from the Josephine pattern and on to the three-lobe tying pattern of Carrick Mat when arranged correctly for use as a bend or as a fixed loop.  This three-lobed pattern may be regarded as a typical, natural knot pattern that, in one case, has complete over-under patterning and no twists between any lines.  This allows for the possibility of other knots that have the same tying outline as the three-lobed presentation of the Carrick Mat but fail the complete over-under patterning and have at least one degenerate twist between two lines.  The Zeppelin Knot is just one such instance.

[DerekSmith]

For all our technological brilliance, the human mind is really quite limited in terms of our spatial perception ability.

We can see and perceive continuously in two dimensions, the 'x' and 'y' flat plane in front of us.  But in contrast, we really are quite limited in our perception of the third dimension of depth.  This is essentially rendered as layers of 'this is in front of that' or 'that is behind the other'.  We find it very hard to perceive a continuum in the third 'z' dimension.  If we need to study the continuum of something in the 'z' dimension, we turn it around or turn around ourselves to reorientate the 'z' dimension to become an 'x' or 'y' dimension.

But knots are obligate three dimensional force vector objects, and our minds have immense problems perceiving them in their functional form.

Consequently, we fall back on pictogram tricks, visual mnemonics that allow us to remember a structure in two dimensions (well, two dimensions plus over and under) that will hopefully transform into the functional knot we desire.  But these are just tricks to compensate for the limitations of our minds - much like the childhood rhyme 'Richard Of York ...' to help them remember the sequence of colours in the rainbow.

But for all the limitations of our mind's abilities, we should at least acknowledge our use of visual mnemonics and recognise that these are pictograms - they are not knots - they are convenient memory aids that work well in the limited spatial awareness capacity of our limited minds.

The Carrick and the Zeppelin bends are two exquisite knots.  Yes, we can pull them open and lay them out into memorable two and a half dimensional pictograms that we can perceive and remember.  The knots are all the more impressive in that once we start to apply force to the loaded legs, they will uniformly morph to form the cogging lock, then without any need to dress the knot, they will transform from the pictogram into the functional knot.

But the pictograms are not knots and it is almost impossible to perceive and predict how the pictogram form will deform and behave when loaded.

A few years ago now, I started out attempting to describe a knot structure with a shorthand called the 'Overs Index'.  How wrong I was...  Over the period of about three months I came to realise that I was preaching a nonsense and that the only way of understanding a knot was to attempt to comprehend its fully 3D force vector components, and from this, attempt to understand the interaction of those components within the specific knot.

Perhaps understandably, of all the components so far identified, the Carrick Component is by far my favourite and can be used as the heart of a great many well functioning knots, although very few of these function as well as the knot formed by interlocking two Carrick Components - the Carrick bend...

[xarax]

the knot formed by interlocking two Carrick Components - the Carrick bend...

   Two interlocking Carrick components form two knots, not one : There are two Carrick bends, depending on the way their tails cross each other ( just as there are two Ashley ABoK#1452 bends ), because, although their topology is identical, their geometry is very different. If we could see the microscopic deformations of the structure of the material, and "watch" the "flow" of the tensile "forces" within the rope, we would never characterize those two different knots as one. An indication about the two completely different ways the Carrick components lock to each other in each of those two knots, is provided by the visible deformations of the cross section of the corresponding segments of the ropes, which occur in different places in the two nubs.
    Of course, I am sure that knot tyers will continue to speak about ONE Carrick bend, for some decades, at least, more !  Regarding knots, the situation is not very different of what is happening in many other fields : the new generation of knot tyers will be able to see the obvious, only after the old generation leaves the stage. ( This is what had happened in the case of the 19th century physicists and the quantum theory, for example ).
   The interested reader may wish to have a look at the "second" Carrick bend - and may even wish to actually tie it, and "see" how different it is from the "first" one. Or, he may ignore this post, too, and keep believing what suits to the time-honoured practice of knotting : repetition, regardless of the truth or practical value of what is repeated.
   

[DerekSmith]

Hi Xarax,

As you seem interested in one of my favourite components, lets get a little bit picky shall we.

For the sake of clarity for other readers, I think what Xarax meant when he said that two interlocking Carrick components forms two knots, was that two interlocking Carrick components forms one knot that can have one of two forms.  One where the SPs are diametrically opposite in the Carrick matt, and the other where they are on an adjacent side of the matt.  And if you only ever tie the Carrick starting from the matt 'pictogram', you will always tie one or the other of these two forms.  In that respect Xarax, you were right.

But...  The Carrick component is not symmetrical - it is 'handed'.  That is, you can make it in either a left handed or a right handed form, where the WP lays either to the lhs or to the rhs of the SP.  With two 'hands' of the Carrick component, these can be combined in four ways i.e. lhs + lhs,  lhs + rhs, rhs + lhs and finally rhs + rhs.  One of these produces the 'Opposite' form of the knot, one forms the 'Adjacent' form and two of them form a perfectly usable Carrick bend which surprisingly can be morphed back into the Josephine matt plus a twist.  One is opposite and the other adjacent.

You can only see subtleties like these if you  go down to the discrete components and then carefully examine ways of combining components into knots.  But this is a new way of looking at knots, and as Xarax stated, it will be some time before a new generation of 'component combiners' replaces the established methods of rote tying from pictograms.

When I tie a Carrick bend or a Carrick loop, I tie two components, I no longer go anywhere near a Carrick matt pictogram.  Although having said that, I will mostly use a slight variant of tying the Chinese Button Knot as a very rapid way of forming a Carrick loop.

Note - I don't know what you had to do to force the Carrick into the structure pictured or why.  One of the joys and advantages of the Carrick is that it is self dressing - just pull the SPs.

[xarax]

two interlocking Carrick components forms one knot that can have one of two forms.

   The "one knot" is an ambiguous term ! "One", yes, but only topologically ! Geometrically, and regarding the actual distribution of the tensile forces within it, TWO ! Not two forms of one knot, TWO KNOTS !  Topology does not determine geometry - as we have see (1), two topologically identical knots can be VEEERY different geometrically and structurally ...That is what is happening with the Pretzel-to-Pretzel bend, and the Hunter s X bend, shown at (1), and with the Ampersand and the Scot s TIB bowlines !

   There is NOTHING "FORCEFUL" that you have to do, to "force"(sic) the Carrick mat to be transformed to the Carrick bend X ( X = crossed tails ). Just cross the tails the other way - and then "just pull the SP s" ! 
   One way to assure oneself that he/she should better talk about a different knot, is to watch how this bend is changing shape while it is loaded - and how different this change is, in comparison to the corresponding change of the "common" Carrick bend.
   I was tempted to propose an altogether new name for this knot (2) - but then, should we do the same for the two, or three, different Ashley s bends ? So, I decided to keep the same generic name - but mention the difference, each and every time one tells something about THE ( supposedly ONE ) Carrick bend, which is TWO !       

When I tie a Carrick bend or a Carrick loop, I tie two components, I no longer go anywhere near a Carrick matt pictogram. 

  Then, you miss the GREAT joy of watching the Carrick mat been folded onto itself, and transformed into one of the two Carrick bends ! And THAT joy is the only reason. IMHO, one would ever chose to tie a Carrick bend - there are dozens of dozens bends out there, many of them being easier to tie, to inspect, and to untie, more symmetric, with tails more aligned to the Standing Ends, and probably stronger.

1. http://igkt.net/sm/index.php?topic=4201.msg25682#msg25682
2. http://igkt.net/sm/index.php?topic=4128   

[DerekSmith]

   The "one knot" is an ambiguous term ! "One", yes, but only topologically ! Geometrically, and regarding the actual distribution of the tensile forces within it, TWO ! Not two forms of one knot, TWO KNOTS !  Topology does not determine geometry - as we have see (1), two topologically identical knots can be VEEERY different geometrically and structurally ...

This must be a failure of the English language.  If I ask three people to tie a square knot and they each tie an identical knot, the the question 'How many knots are there?' the answer of 'One' as in only one type of knot, and 'Three' as in three knots of the same type will both be equally correct.

But : the point in hand was the assertion that two Carrick components could be used to create two forms of Carrick bend.  I contest this and state that two Carrick components can create four forms of Carrick bend.

As for missing 'the GREAT joy of watching the Carrick mat been[sic] folded onto itself' - that is a matter of personal taste and I now get greater satisfaction from creating a knot from its functional components.

But I am intrigued by your claim that there are "dozens of dozens bends out there, many of them being easier to tie, to inspect, and to untie, more symmetric, with tails more aligned to the Standing Ends, and probably stronger".  I wonder would you be willing to cite two or three for me that you feel fit this table of benefits, so that I could compare them for myself?

Derek

Edit   5 May  23:55

Having re-read the previous post, I now see that you meant that two Carrick components can be dressed into (at least) two final geometric styles - the Carrick and the Carrick X.  This of course means that any two Carrick components can be dressed into (at least) 8 knot variants, four of them Carrick variants and four of them Carrick X variants.

Having made this concession, I have removed my posts below as they add no value to the OP

Derek

[Stagehand]

Hello Derek,
     Thanks for the gracious reply.  I hope to have understood your remarks and will try to show some comprehension.  May I say how you see the Carrick? 
     The Carrick Pattern is then the freely ranging workings of Carrick Components that may be combined variously in tying patterns.  Once the Carrick Components are tied in some way, the pattern model must be abandoned in consideration of the dynamic parts of mechanisms under tension as they move and change. 
     If that is about what you mean then I can appreciate the kind of rigor you ask for.  I don't know if I can help with this.
     The Mathematical Knot Tables represent a different kind of rigor.  All real knots, single-line or multiple-line, that have or infer complete circuits are then identifiable uniquely in the tables.  This is as general as identifying a coffee mug with a donut for the holes.  This identifies the Zeppelin Knot as Rolfsen 6_3, a complete knot of six crossings.  More particularly composed of two plain crossings and two twists.  Whether these aspects are understandable physically, they will persist and be always identifiable in the physical knot.
     In this context, the Carrick Pattern can be seen as an ambiguity between two knots identifiable in the Mathematical Knot Tables.  The Same-side Carrick Bend nearly completes and clearly infers the mathematical knot Rolfsen 8_18.  The True Carrick Bend (ABoK #1439) nearly completes and clearly infers Rolfsen 9_40.  This ambiguity is manageable through all real knots where the Carrick Geometry is identifiable mathematically.

[Dan_Lehman]

a fail and degnerate
knots ... that infer ...

??  What language is this --this use of "fail"?
I assume you mean "imply" by "infer"?

The Mathematical Knot Tables represent a different kind of rigor.
All real knots, single-line or multiple-line, that have or [imply]
complete circuits are then identifiable uniquely in the tables.

What do you mean by "real knots"?
--all mathematical knots (such as are
presented in the Rolfsen table)?
or
--all practical knots (such as can be found
in knot books --sometimes only!-- or out
"in the wild.")?

For practical knots --e.g., the bowline & carrick bend--,
how do you convert them into mathematical knots
so that you may identify them in the Rolfsen table?

This identifies the Zeppelin Knot as Rolfsen 6_3,
a complete knot of six crossings.

A zeppelin knot is composed of two, interlocked
overhands, whereas a carrick bend is of two
interlocked crossing knots : one should be surprised
if the former has fewer crossings than the latter!
When I connect ends of a zeppelin knot to convert
it into a closed curve ("complete circuit"), I get a knot
with 11 crossings, not 6!

... the Carrick Pattern can be seen as an ambiguity between two knots
identifiable in the Mathematical Knot Tables.  The Same-side Carrick Bend
nearly completes and clearly [implies] the mathematical knot Rolfsen 8_18.
The True Carrick Bend (ABoK #1439) nearly completes and clearly
[implies] Rolfsen 9_40.

Really?
Again, by what rule do you convert the practical *knot*
into the mathematical one?  We might also ask What
is a carrick bend --the lattice form or the capsized form?
(for it makes a difference where in the Rolfsen table
you end up!)?

This ambiguity is manageable through all real
knots where the Carrick Geometry is identifiable mathematically.

??  How does one "identify the carrick geometry mathematically?"

--dl*
====

[DerekSmith]

Hi Stagehand,

Thanks for coming back into the discussion.  Yes, your are quite correct, I was arguing for a knot to be perceived as a physical analogue 3D force vector machine.

Although I accept your argument that some element of knot structure can be captured in the Rolfsen simplification, and that this description can be utilised to differentiate some knots from others (as in the case of the very similar Overs Index), the fact remains that in order to arrive at the Rolfsen value, a massive amount of simplification must be performed and a staggering amount of information must be discarded, to such a degree as to render the Rolfsen value to be virtually meaningless in the working world of knots (i.e. 3D force vector machines).

I would use the simple demonstration of this by taking the wonderfully functional Carrick bend and re-dressing it into a 'contortion' around a straight cord where the contortion can simply slide right off the cord - no longer a functional bend.  The Rolfsen values for the contortion on the straight cord and the highly functional Carrick bend are identical, yet in the 'real' world, one is a knot and one is dysfunctional tangle.  In order to arrive at the Rolfsen simplification, all the 3D force vectors, the frictional coefficient, the cogging and torsional aspects, let alone the physical aspects of the cordage involved, have had to be discarded.  Yet it is within this rich matrix of variables, that the real world of functional force vector machines operate, and for some of us, the real inspiration and pleasure of knots exists.

Derek

[Dan_Lehman]

and re-dressing it into a 'contortion' around a straight cord

In case anyone didn't understand this,
what he's saying is to straighten one of
the joined pieces so that the other has
all of the *knotting* in it.

But, to my question above, first one must
explain the rule/method by which the practical
knot is *closed* to become the mathematical
one --this is the question : how so ?!


--dl*
====

[DerekSmith]

In case anyone didn't understand this,
what he's saying is to straighten one of
the joined pieces so that the other has
all of the *knotting* in it.

--dl*
====

Nicely put Dan.  Thank you.

As for your question, I had exactly the same problem when I started applying the Overs Index to bends.  The rule one decides upon can impart a new (totally fictitious) 'engagement', purely for the sake of creating the 'mathematical knot' needed to apply the analysis.  But, the reality remains that the bend is in fact two (non)knots, interacting to transfer the forces delivered by one, into the other. 

In order to describe/define the bend, each component needs to be described, along with the way in which each one interacts with the other.  As no force is transferred through the WEs, these cannot be part of the knot description, so any rule that forces them to be part of the description has to be a falsification of reality.

Derek

[xarax]

one must explain the rule/method by which the practical knot is *closed* to become the mathematical one --this is the question : how so ?!

  This is a common mistake knot tyers are doing all the time : We are separating the mathematical from the "practical" knots, in a wrong way : as if a "practical" knot, when it is "closed", i.e., when its two ends are fused together, so there remains none, would be transformed into a mathematical knot ! 
 
  First : The "opposite" of a mathematical knot, is a physical knot ( = a macroscopic knot in classic 3D space, made from "atoms" that are remain at a certain distance to each other due to "bonds", of electromagnetic origin. I do not know what happens inside the nucleus of an atom, where the binding force is the strong nuclear force, or in between the stars and the galaxies, where the binding force is the gravitational force.  There, any "strings" that may exist can, perhaps, penetrate and go through each other, unlike a "string" made by chemical atoms, so I guess there can be no "knots" like the ones we know...)
   Some of the physical knots, usually the most simple ones, happen to be "practical" knots as well - the distinction between a practical and a not-practical knot is not so clear, as we all know ! 
   There is a branch of applied mathematics, which studies the so-called "ideal" knots, which are mathematical knots endowed with geometrical properties as well - but deprived of any other physical properties of materials, as friction, temperature, elasticity, etc. "Ideal knots" pose very difficult geometrical problems, and we may say that none of the most essential of them has been solved : for example, we do not know the exact mathematical equation the path of the lines follow in even the most simple compact closed ideal knots ! With the advent of computers, ideal knots can be studied by approximate simulations, which, although not exact, offer nevertheless beautiful mental and visual images of entangled ideal knots :
    http://igkt.net/sm/index.php?topic=3728
   
  Second : The "opposite" of an "open" knot, is a "closed" knot. There can be "open" = two ends knots, and "closed" = no ends knots, in mathematics as well as in the 3D physical space. However, in the special branch of mathematics which studies the topological, only, properties of mathematical knots, ( called " Knot theory" ), an "open" knot, alone, is of no much interest.
   Now, in mathematical as well as in 3D classical physical space, "closed" knots can stand alone, or be entangled with other "closed" knots. In mathematical knots, such topologically entangled closed knots are called "links". Knot theory studies links, and there are tables of links as there are tables of ( closed ) knots :
   http://katlas.math.toronto.edu/wiki/The_Thistlethwaite_Link_Table

  Third : when we like to find the "corresponding", to a physical, mathematical knot, we have to pair apples to apples : If we have a stopper, we have to search for a not-linked closed mathematical knot, and just to "cut" it somewhere. Where exactly, it does not matter, as a mathematical knot has no geometrical properties : all we can study in it, is its topological properties. If we have an end-to-end / bend, or a eye-knot / loop, we have to search for a two-link mathematical knot, and just "cut" both links, in the case of bends, or the one link, in the case of loops.
   This "cut" should be interpreted in two ways : First, it is a cut which changes the topology of the mathematical knot from a "closed" to an "open" one. Second, the newly generated, by this cut, "ends", should be considered as "points" been transported to infinity, that is, not accessible to any further manipulation - just like the Standing and Tail Ends of a TIB loop, for example. ( The Tail End of a finished bend is also such a point, but, as it is physically accessible to us, we tend to consider it somehow differently ). 
   
   Some people follow a different approach : To find the "corresponding" closed mathematical knot to a bend, they find which not-linked, closed mathematical knot can be "cut" twice, in two "points", and be transformed in two pseudo-entangled open knots. However, this happens only because they do not have access in tables of links - and because there are no tables of symmetric two-part links ! The mere enumeration of the symmetric links, "corresponding" to symmetric bends, ( where two or more closed mathematical knots are topologically entangled together ) is a difficult thing, as far as I know...
   I would be glad to learn more things about symmetric two-"knot" links, as they are of much interest to us.
   For a recent thread about simple mathematical knots  ( up to 9 crossings ) "corresponding" to simple stoppers, see :
    http://igkt.net/sm/index.php?topic=4822
   I would be also veeery glad, if somebody does the same thing for the links "corresponding" to the known bends...before the end of this century.         

[Dan_Lehman]

There needs to be a separate thread in this
very (sub-)forum (i.e., "Knotting Concepts...")
regarding operations on knots, such as match
terms "reverse", "corresponding {end-2-end/eye/...}",
and in matching some "practical"/"physical" knot
to one that lies in "mathematical knot" tables.
(One might realize that matching all TIB eyenots
to Rolfsen_0_1 is not terribly helpful!  Maybe there
needs to be a special rule for such cases,
to gain meaningful/varied results!?)

In the immediate case, we are told that certain
physical knots match particular knots in the Rolfsen
Table : I want to know by what method one can
make this match.  (And I should remark that I'm
quite surprised that X. has been completely quiet
about the assertion that his beloved, worshipped,   
zeppelin ~= Rolfsen_6_3 !?)

--dl*
====

[xarax]

   As I had mentioned in my previous post, when we want to "match" a physical/practical end-to-end knot ( as the Zeppelin bend, for example ), we should NOT use the Rolfsen Table, but the Thistlethwaite Table of mathematical links ( up to 13 crossings )(1) - because, as common sense tells us, a two-link knot "corresponds" to a two-link knot !     

   See the attached pictures for some symmetric links with 10 crossings, from another source (2). To which physical/practical knots they correspond, I leave it as a knotting exercise !     
 
(1)  http://katlas.math.toronto.edu/wiki/The_Thistlethwaite_Link_Table
(2)  http://www.indiana.edu/~linkinfo/

[xarax]

   Some more symmetric links. ( I do not know where one can find a table of all simple symmetric links, "corresponding" to the linked knots with the number of crossings of the physical/practical end-to-end knots / bends we know/use...)