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).
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 ! (*)
(*) 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 …
[Again, I import important matter from another thread
over to this one, where it gets fuller attention, and is
right on the topic.]
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)?)
[quote="Dan_Lehman post:16, topic:5304"]
how does your common sense deal with a >>hitch<< --that thing that has (let us say) a rigid, non-cordage object in its midst!?
[/quote]
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 [i]hitches[/i], 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.
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 !
(*)