A question of names. Knots based upon/resembling the fig.8 knot

[xarax]

   (See the attached pictures and labels.) A suggestion for decriptive names of (a series of) knots based upon, and somehow resembling, the fig.8 knot.


   

[Dan_Lehman]

A suggestion for decriptive names of (a series of) knots based upon,
and somehow resembling, the fig.8 knot.

The fig.8 knot --as it's commonly formed(!), that is--
comes but in one position in what can be seen as a much
richer series of *knots* than would owe allegiance to that
common name.  The immediate *successor* which you
show has a common name amongst kernmantle-rope
users as the "fig.9" --taking "8" as the base to which
one "adds" a (half) turn.

verbal illustration of this series's other forms
Consider these two knots as though they are dancers,
and their loops represent the dancer's arms conjoined
and holding one leg high above head; now let that leg
come back down, bringing the arms with it, to the crotch
--and you have a symmetric knot (series) with ends
coming in to twist up to a point of departure in which
they arc in opposite directions to join at the crotch.
That is one form, which I see beginning not with the
fig.8 but with the overhand and continuing
infinitely.

There is another symmetric form in which the crossing
of parts occurs on either side of a void in the center,
until some interlocking at the top (vis-a-vis my "dancer"
orientation of things).  Ashley's #521 is the fig.9 member
of this series (and makes an interesting mid-line stopper,
with careful dressing & setting!).  I think that both the
fig.8 & overhand either have a common form for
these symmetric series, or some perhaps dubiously distinct
via tricky dressing forms --they lack sufficient twists to get
into trouble in two directions, so to speak!   

So, I beg off the latter knots shown here, though they
might belong to some series that sees the fig.8 as a
("degenerate" --a formal term) predecessor!?


Btw, although I know that the different forms of
the knots exist, I found it extremely difficult to move
from the common fig.9 form to the symmetric ones,
for some time (finally, I figured the sort of "dancer" view
and its transition; getting to #525 is still more of a struggle!).
I cannot move (yet?) from one to the other symmetric form
--without reaching the asymmetric form and working from that.


--dl*
====

[xarax]

   Thank you Dan Lehman,

   It is amusing that Ashley himself did nt find a better name for the "Intermediate"#521 ! That proves that the issue is of some interest, because the transformation of the one knot to any other, belonging to a coherent series of related knots, is not self evident.
   The "twist"or "untwist" term I propose is quite descriptive, and corresponds to a possible actual way we can go from the previous knot to the next one. I have used the same term where the standing ends and/or tails of a knot make an additional turn around a strand, that was not present in the previous, simpler knot. ( See, for example, the "twisted" Hunter s bends). Here, I think that it is even more appropriate, because it also means the 'twist"or "twist" of one fig . 8 s bight - or a bight that has been generated by a previous such twist, that also was a product of a "twist"or "untwist"of an initial fig. 8 s bight. Pulling the tail out of its final tuck, twist or untwist the bight from which it was going through previously, and then re-tuck it again through the same side of the new, twisted or untwisted bight...It is a simple, hard to confuse or forget operation.
   The "fig. 9" name leave me cool, because I see no figure 9 shape there, and because, when I do wish I see a shape of a digit, I actually see TWO figure 9 shapes  ,  placed point-symmetrically the one to the other. It is a wrong  name, a wrong picture, it leads to the silly suggestion that the next knot should be named as "fig. 10', and the next as "fig. 11", and so on

  I think that both the fig.8 & overhand either have a common form for
these symmetric series, or some perhaps dubiously distinctvia tricky dressing forms

I have tried to keep the two, most basic and topologically so different forms, the overhand and the fig. 8, separated - for the time being - , and to see them as the primitive ancestors of two corresponding series. Perhaps this is not so general or wise a scheme, but I thing I should finish this modest plan first, and only then see how those two distinct series are related - because they are related, for sure, as every knot is related to every one else.

[Dan_Lehman]

The "twist"or "untwist" term I propose is quite descriptive,

I'll leave off any thoughts about the degree you take this,
in terms of particular direction; but I want to say that I
regard a "twist" as involving two strands (or more) equally,
in contrast to some structures that have what I call "wraps"
in which one strand goes around another, straight strand
--which state might result from an imbalanced loading of
some "twisted" strands.  But, again, ... just to say, for me,
difference between "twist" & "wrap".

The "fig. 9" name leave me cool, because I see no figure 9 shape there,
 and because, when I do wish I see a shape of a digit, I actually see TWO figure 9 shapes  , ...
It is a wrong  name, a wrong picture, it leads to the silly suggestion that
the next knot should be named as "fig. 10', and the next as "fig. 11", and so on ...

Bingo!  This is precisely what IS done (though, for practical
purposes, I am unaware of any name >10 --the Fig.10 was
tested in a 2001 Lyon Equip. HSE report, and found to be
stronger (eye knot) than the Fig.9 & 8, slightly).  I quite
concur in your critique of this nomenclature, BUT I readily use
it for communication --because it succeeds.  I might also note
that I can refer to "fig.8" in orientations that don't match the
common one, as done above.  In regard to this series, note per
my observations, one has a variety of geometries (hence, "figures")
that each topological structure can assume.  Perhaps there will
be some canonical form decided upon (one with easy continuation
by adding a (half-)twist) and a name generated for that, with
then the derivative forms referred to by some derived or other
name.
In the quick, the "fig.x" nomenclature at least can be readily
comprehended, with its simple well-known base and the
easily counted (half-)twist additions.  (We can use the term
while holding our noses, and awaiting improvement.      )

  I think that both the fig.8 & overhand either have a common form for
these symmetric series, or some perhaps dubiously distinctvia tricky dressing forms

I have tried to keep the two, most basic and topologically so different forms,
the overhand and the fig. 8, separated --for the time being-- ,
and to see them as the primitive ancestors of two corresponding series.
Perhaps this is not so general or wise a scheme, but I thing I should finish
this modest plan first, and only then see how those two distinct series are
related --because they are related, for sure, as every knot is related to every one else.

Well, you have also seen as, called "eights" and maybe overhand
"pretzels" with knots that only resemble these in silhouette
--which can be a quite *real*/effective similarity, in terms of having
"loops" to surround/nip/contain, and so on,
but which are not related in the way the "fig.x+n" series is.


--dl*
====

[xarax]

The "twist"or "untwist" term I propose is quite descriptive,

   (I'll leave off any thoughts about the degree you take this, in terms of particular direction;)
   I regard a "twist" as involving two strands (or more) equally, in contrast to some structures that have what I call "wraps" in which one strand goes around another, straight strand
...difference between "twist" & "wrap".

  (I will not insist in the "particular direction" issue. Whether something is a "twisting", or an "untwisting", can be seen after one actually repeats the operation a few times.)
   I wanted to use the same term, to describe both things :
   1. The operation with which we transform one knot into another - by removing one (or both) tail(s) from the last bight(s) they go through , "twisting"or "untwisting" this/those bight(s) around its/their node(s) - the point where the one leg of the bight meets the other - and then re-tucking the tail(s) through the same but now "twisted" or "untwisted" bight(s) again, following the same direction as previously.
   2. The form of the two limbs of the bights going around each other, in the way the standing ends and/or tails are 'twisted' in the "Twisted Hunter s bend", for example.
   Is there a more appropriate word than can describe both those two things ?
   ( I have already accepted the usefull distinction between the "half-twisted" (180 degrees) and "twisted" (360 degrees) rotation of the bights around their nodes and symmetry axeses.)

Well, you have also seen as, called "eights" and maybe overhand "pretzels" with knots that only resemble these in silhouette--which can be a quite *real*/effective similarity, in terms of having "loops" to surround/nip/contain, and so on, but which are not related in the way the "fig.x+n" series is.

   All very true. See the attached pictures for the same operation, now on the overhand knot ( the symmetric silhouette of it... ). It is difficult to distinguish the two series, if we do not notice that, at the fig. 8 series, the tails leave the symmetric loose knot from different sides of the bights, while at the overhand knot series they leave from the same side.

[Dan_Lehman]

Well, you have also seen as, called "eights" and maybe overhand "pretzels" with knots that only resemble these in silhouette--which can be a quite *real*/effective similarity, in terms of having "loops" to surround/nip/contain, and so on, but which are not related in the way the "fig.x+n" series is.

   All very true. See the attached pictures for the same operation, now on the overhand knot ( the symmetric silhouette of it... ). It is difficult to distinguish the two series, if we do not notice that, at the fig. 8 series, the tails leave the symmetric loose knot from different sides of the bights, while at the overhand knot series they leave from the same side.

Whoa!  You have this backwards --you show here,
as the first knot, exactly a (n oddly shaped) fig.8
in contrast to the fig.9 in the OP, which is a full turn
*beyond* the overhand !!

--dl*
====

[xarax]

Whoa!  You have this backwards --you show here,as the first knot, exactly a fig.8

Exactly ! A one bight, half-twisted overhand knot is a fig.8 knot. This is an example of the transformation I was talking about. We have to be able see the (many) individual knots as (one) generic/parent knot, that is transformed by similar operations.
Does this has any relation with the "dancers" picture you have in mind ?

[Dan_Lehman]

Whoa!  You have this backwards --you show here,as the first knot, exactly a fig.8

Exactly ! A one bight, half-twisted overhand knot is a fig.8 knot. This is an example of the transformation I was talking about. We have to be able see the (many) individual knots as (one) generic/parent knot, that is transformed by similar operations.

I'm afraid I don't see any "overhand" here,
and that something else (truth) is being twisted.
But, OTOH, "yes", from the **pretzel** geometry,
with the overhand's "twist" being merely (we
might posit, degenerately?) a crossing, then one
(half-)twist of that begets the fig.8, and another
the fig.9, and so on.

It's that presenting the asymmetric form with but
a changed shape that doesn't *hold* (in contrast
to the center-twist series with a "pretzel" beginning,
where the edges of the pretzel can be seen to hold
back ends wanting to UNtwist).

A similar sort of deceptive crossing occurs in some
drawings of this series I have (will try to photo-post)
where each silhouette/shadow image stands for a
pair of series members (distinguished by how
one orders the crossings, brought out of shadow).

(The actual, dress-&-set-able *knots* that result
from the first couple members are interesting and
potentially useful; I don't know about ones of "10"
and beyond, really --other than the asymmetric form
where one can see the wrapping/twisting as making
some benefit to *static* strength (of dubious gain).)

--dl*
====

[xarax]

 

I'm afraid I don't see any "overhand" here, and that something else (truth) is being twisted.

  I have thought for a while what should be the proper response to this ... , but, after a while, the Gandhi-like person in me prevailed, ,  and I came up with a plausible explanation, that forgives you ! 
  You have lost your glasses in this bicycle accident of yours, and now you are saving money to buy new - or you try to train your eyes to see without glasses...( like I am always trying to do... ). So you do not "see" very well...OK.
   ( However, when we do not "see", we do not accuse people that they are twisting the truth, because this is only an indirect way to tell them they are liars, or crazy...We just make a polite question, like this :
   "I am afraid I do not understand what you are trying to tell me here...Could you, please, repeat it in other words, show me some other pictures, etc.....?")
   I have already described, in a detailed way, the operation that transforms the knots here;  It is the same operation that generates the knots shown in Reply#1, starting - or ending - from/at the fig.8 knot, and the knots shown at Reply# 4, starting - or ending - from/at the overhand knot.

  1. The operation with which we transform one knot into another - by removing one (or both) tail(s) from the last bight(s) they go through , "twisting"or "untwisting" this/those bight(s) around its/their node(s) - the point where the one leg of this bight meets the other - and then re-tucking the tail(s) through the same but now "twisted" or "untwisted" bight(s) again, following the same direction as previously.

   So, let me try again: "See"  the attached pictures : You have not seen the overhand knot, but now you see it...
   Picture 1 : An overhand knot !
   Picture 2 : Untuck the tail from the last bight. Here we have a "one bight" operation, so we untuck only the "upper" tail from the "lower" bight. In the "both bights" operation, we untuck both tails from their last bights.
   Picture 3. Twist  the ("lower") bight 180 degrees ( half-twist), around its node - the point where the one leg meets the other. I hope it is easily understood what is "twisting" and what would be "untwisting" - as it happens in other cases.
   Picture 4. Tuck the tail again, through the same bight, following the same direction as previously.
  ( Picture 1 and Picture 4 are pictures of the same knot, the overhand knot, the first (Picture 1) not twisted, the second (Picture 4) "one bight, half-twisted", by the operation described abve... as everyody can easily see.)

[Dan_Lehman]

I have set out the tangles that I see as constituting
a series which is in my mind in responses here, in

http://igkt.net/sm/index.php?topic=3838.0


--dl*
====