Establishing Operations (& Terms) for Knotting

[Dan_Lehman]

The field of knot theory (topology) has some defined
terms for relations between knots.  It should be of
help in practical knotting to establish a set of such
operations and terms for them that enable the
field to be explored and better explained & understood.

E.g., "reverse" is a term that should denote some
well-defined operation that one can perform at
least on some physical knots --such as loading
the tail vice SPart of a stopper,
loading the tail opposite the eye of an eyeknot(?!)
loading the tails vice the SParts of an end-2-end knot(?!).

There are particular correspondences between
eyeknots & end-2-end knots.  A commonly
understood one is to begin with the latter
and connect (conceptually) one tail with
one SPart to get the corresponding former
(e.g., the sheet bend & bowline have such a
correspondence; but note that given the
asymmetry of the former, there is a 2nd
such correspondence that could be made!).

... and so on.

As I have raised today in another thread,
we should have some method for matching
physical knots into the "closed curve" forms
of knots in (mathematical) knot tables (which
thus have no ends, only entanglement, and
listed by the number of (minimal) crossings).

--dl*
====

[Dan_Lehman]

I'm re-posting here the complete post by Xarax
in another thread, which addresses this topic.

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 one 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.       

--dl*
====

[xarax]

   Miles uses the term "reverse bend" - and he states that : " mostly the geometry of the knotted part of the reverse bend coincides with that of the original bend... but sometimes not" ( p. 30) -  and he offers the example of ABoK#1424 / M A11. He could also had used the example of two "reverse", to each other, but quite different bends, which himself had noticed; the ABoK#1422 and the Violin bend. (0)

   I do not understand this... The geometry of the knot is not determined uniquely by its topology - for the same topology, and the same Standing and Tail Ends, we can often tie more than one stable knots (1)(2). Moreover, the geometry of the "reverse knot" does NEVER coincide with that of the original bend ! A different loading, leads necessarily to a different stable form of the structure, and a different geometry : so, not only "NOT-mostly", but "never" !

   Regarding the different loadings and geometries of a bend turned into a loop, the ABoK#1424 is, again, a good example, as it generates two quite different knots :
   http://igkt.net/sm/index.php?topic=4452
   
   Regarding the case of the eye-knots, I believe we can extend the meaning of the "reversing" operation, and call any pair of loops where the eye legs and the Standing and Tail ends have been swapped, "reverse" loops. However, as the Standing and the Tail Ends can also be swapped, any loop has two "reverse" loops - which poses yet another problem of knotting nomenclature !  (*)

0. http://igkt.net/sm/index.php?topic=3939
1. http://igkt.net/sm/index.php?topic=4201
2. http://igkt.net/sm/index.php?topic=4877.msg31925#msg31925

(*) P.S. Perhaps the "reverse" of an eye-knot, should be the one eye-knot where : eye-leg of the Standing End <==> Tail End, and :  eye leg of the Tail End <==> Standing End  - i.e., where the "reversing" operation has been applied on both properties, " in which of the two sides of the nub the eye legs are " , AND : " in which of the two continuations of the eye-legs the Standing End and the Tail End are ". So, for the other eye-knot, where : eye leg of the Standing End <==> Standing End, and : eye leg of the Tail <==> Tail, we can use the noun "reverse" with a prefix, like "para-reverse", or something ...

[Dan_Lehman]

[Again, I import important matter from another thread
over to this one, where it gets fuller attention, and is
right on the topic.]

What does common sense lead you to match for an eyeknot (e.g., the bowline

  The nub of an eye-knot, in general, and a bowline, in particular, is nothing but a link,
where each part is "cut" in two points, just as it happens with end-to-end knots.
The only difference is that the "corresponding" ( to the mathematical knot ) physical/practical knot
is loaded only by the three from its four limbs.  However, I should repeat that, in Knot theory
( so, not in the theory of Ideal knots, which are also abstract/mathematical knots ),
the only property a knot has is its topology.
We should not get confused and interpret the representations of mathematical knots
studied in Knot theory as having any geometrical significance : they are not telling anything
about the knot's geometry, because, in Knot Theory, there is none : the only thing there is,
is an abstract continuous line, which can have any shape or length, provided it does not
penetrate/go through/cut itself or any other.
   So, a PET eyeknot is just a link where the one of the two "cut" lines,
which corresponds to the nipping structure, is just a "straight" or "curved",
but un-knotted nevertheless, continous line, from a point at infinity to a point at infinity,
i.e., it stretches between two points which are not accessible and can not be manipulated
any further.  The corresponding of an overhand-knot based eye-knot,
as the infamous so-called "Zeppelin" loop, is a mathematical knot where the one link
is topologically equivalent to the overhand knot, and not to the unknot,
so it can not be represented as a "straight" line.

What does common sense lead you to match for a ... stopper knot (e.g., Ashley's stopper)?

  Ah, I believe that, in this issue, your common sense, too, is enough ! 
  A stopper is topologically equivalent to one mathematical knot, of the many shown in the Rolfsen table ( so, NOT is a table of mathematical links - it is un-linked ). If you search in the Forum, you may be able to discover a thread dealing with fig.9 stoppers - but you can also press the proper key on your keyboard, and select it from the reference below :
  http://igkt.net/sm/index.php?topic=4822

Hmmm, that's interesting.  By some other consideration,
Ashley's ("Oysterman's") stopper ~= bowline
--collapse the eyeknot's eye, load the eyeknot's tail.
Above, you in some way resist/prevent this
by holding the *ends* away --eyeleg & SPart, e. & tail--,
which greatly restricts manipulation.
(And maybe this points to Conway diagrams (IIRC)?)

how does your common sense deal with a >>hitch<< --that thing that has (let us say) a rigid, non-cordage object in its midst!?

   Here we go again...
How easily a property that is "physical", i.e., belongs to the one part of the "correspondence" relation,
which concerns the physical/practical knots, jumps on the other part,
which concerns the mathematical knots studied by Knot theory, where only topology matters !
   Rigidity, elasticity, stiffness, etc, are properties which are not studied in Knot theory,
so they are completely irrelevant regarding the "correspondence" we are looking for.

Except that in knottable material --excluding rigid objects,
at least, thus--, one can to those topological manipulations!

The hitched object can be rigid or not, hot or cold, stiff or soft, vibrating or not,
clean or dirty, transparent or opaque, etc...It does not matter.
The only thing it matters is its topology after the "cut" --which, for a hitched pole
or ring or a single line or a bight, is an un-knotted continuous line.
   Now, even if we study a hitch as an "ideal" knot, again the "material" properties
of the hitched object play no role whatsoever --but the geometry does.
So, a hitch around a cylindrical object may "fold" and "close" differently than around
an object of a square cross section. I think that nobody has ever studied an ideal hitch ... .
However, an ideal hitch around a pole and an ideal hitch around a rope of the same cross section,
is the same thing :
even in ideal knots, there is no concern about anything other than purely geometrical properties.
   Even if we remain firmly into our realm, of the physical/practical knot,
the rigidity or not of the hitched object plays a minor only role.
It is true that the best hitches around ropes ( the rat-tail-stopper and the climbing gripping hitches ),
are not as good as the best hitches around poles ( the TackleClamp hitch, and all hitches
that utilize a mechanical advantage and a "locking" of the Standing Part mechanism ),
but that does not mean that they are not both hitches, which are represented in the same way.
To my eyes, a hitch within a bight of a rope and a hitch within a ring are the same "thing"
--only some hitches work better in the former and some work better in the later case,
because of some more subtle reasons, which nevertheless cannot/ should-not force us
to use different WORDS for them.
   So, yes, in the case of physical/practical knots, the hitched object is part of the knot :
consequently, the "corresponding" mathematical knots studied by Knot Theory are the links
--a hitch is, for that matter, topologically equivalent to the nub of a ( A, according to Miles )
bend or a ( PET ) loop - a common bowline, for example.

--dl*
====

[xarax]

By some other consideration,Ashley's ("Oysterman's") stopper ~= bowline --collapse the eyeknot's eye, load the eyeknot's tail.
Above, you ...  resist/prevent this by holding the *ends* away --eyeleg & SPart, e. & tail--,
which greatly restricts manipulation.

   We shall NEVER change the most fundamental property of any knot, mathematical or physical : its topology !
   The continuity of the line is a sacred thing - a knot can remain a knot, even if anything else changes, or even disappears - but not the continuity of its line !
   Indeed, I believe that ( the nub of ) an eye knot is a 3-loaded out of the 4-existing limbs of a two-parts/lines knot, a link.
   So, it is not a one-part/line knot - as it looks when we see it as a whole, with the bight of the eye attached on it.
   Topologically, an eye-knot  / loop does not differ from a end-to-end knot / bend - so the "corresponding" mathematical link of the so-called "Zeppelin loop ", and the "corresponding" mathematical link of the genuine Zeppelin knot, the Zeppelin bend, are identical, indeed. That DOES NOT mean, of course, that the "corresponding" ( to this mathematical link ) physical knots, are the same ! First, because topology does not determines geometry uniquely, so two knots that are topologically equivalent, can be veeery different geometrically ! Second, because the loadings of two "similar", geometrically, knots, can be different, so the knots "fold", "close" and settle in different final forms = they become different knots. Loading changes geometry, and it is geometry which determines the identity, as an entity, a "word", of a physical knot. Other things play also a great role : material, construction of the rope, water or ice inside the rope, history/fatique, etc - but it is the geometrical properties that make us tell : "This is an overhand knot", and not anything else.
  So, regarding different loadings, in the case of the eye-knot, it is as if the one Tail End of the corresponding bend has been transformed into another Standing End, and which is now loaded. This changes the loading of the knot, so it changes its geometry, too, but not its topology : the new knot is, as a physical knot, something new, but as a mathematical knot, it remains the same.
   There are many ways to REPRESENT the same mathematical knot, but we should not be confused by the different shapes : the geometry of the diagrams has no relation whatsoever with anything belonging to the knot - but only with our convenience to "read" the information about the topology of the knot which this representation contains. As shown in the thread about the fig.9 knots, it just happens that one particular representation of mathematical "two-bridge braids" or "rational knots" ( the Chebyshef  diagrams ), which has nothing to do with the actual geometry of the "corresponding" physical stoppers, resembles, somehow, the image of loose stoppers. This is useful as a mnemonic aid to us, because we can use a Chebyshef diagram as a tying diagram, to tie the "corresponding" physical stopper easily - but it is not meant to represent the geometry of the represented mathematical knot, because mathematical knots studied by Knot Theory have no geometry !
   Let us imagine a situation where we decide to "cut" a mathematical knot once, in one point, to make two ends, the Standing and the Tail End, but not twice, to make two eye legs. Then, we can "pull" any bight, out of this "open" mathematical knot, and consider that it as corresponding to the eye of an eye-knot. Now, the manipulation is not "greatly restricted", but it is "too-greatly free" !    ANY of the many "segments" between two crossing points can be considered as possible "corresponding" segment of the bight of the eye, so, starting from a certain diagram of an "once-cut" mathematical knot, we can get maaany eye-knots ! The ambiguity we already had, because a certain topology does not lead to a certain geometry, and a different loading of a certain geometry leads to a different geometry, is multiplied a lot !
   Noope, I do not believe that we gain anything if we start from an eye-knot, shrink the eye, and then search the Rolfsen Table for the mathematical "once-cut" knot which corresponds to it - because the opposite leads to too many knots. "Correspondence" may not be "one-to-one", but it should not be "one-to-too-many" either !