Author Topic: The Zeppelin Knot is identifiable as a fail and degenerate Carrick Mat  (Read 15479 times)

Dan_Lehman

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   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 !  :) 

What does common sense lead you to match
for an eyeknot (e.g., the bowline),
and for an stopper knot (e.g., Ashley's stopper)?

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

--dl*
====

Stagehand

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Thanks for the fusillade Dan.  I will be responsive, I may step back.
I used "a Fail" as term of art for a typical tying diagram that might have over-under patterning but then does not.  A precise definition is now limited to the scope of the Rolfsen Knot Table.  A Fail is a mathematical knot presentation (or tying diagram) that cannot be a prime knot presentation because it does not have complete over-under patterning and does not have the minimum or prime number of knot crossings.
At this time I will volunteer that the word 'degeneracy' is used correctly for geometry and that a degeneracy is a twist between two lines.  It follows that for all knots that imply mathematical knots:  If its a prime knot presentation and it has no twists, then it is a Turks-head knot.  This shows how knots are polyhedral but Turks-head knots are polyhedrons (polyhedra).
Thanks for the schooling on inference and implication.  It felt great.

xarax

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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

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. 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 can not/ 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.
   
« Last Edit: May 15, 2014, 02:08:38 PM by xarax »
This is not a knot.

Dan_Lehman

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I used "a Fail" as term of art for a typical tying diagram that might have over-under patterning
but then does not.  A precise definition is now limited to the scope of the Rolfsen Knot Table.
A Fail is a mathematical knot presentation (or tying diagram) that cannot be
a prime knot presentation because it does not have complete over-under patterning
and does not have the minimum or prime number of knot crossings.
I don't find this sense of "fail" defined, even
on-line (where one might expect such jargon).
"degenerate" is a term I understand from logic.

"prime knot" & "prime number" I suppose are
to apply to the same thing, roughly --that the
former has the latter quality (or crossings)?
(But "p. number" runs afoul of the common
meaning in math, of course!)

And "complete over & under crossings" points
to diagrams just so?  But I underline/emphasize
this term because there is at least one (on quick
check) diagram in the Rolfsen Table that lacks
this --it has over->over->under->under crossings
(8_21) : is it still "prime", or a "fail"?!

Quote
At this time I will volunteer that the word 'degeneracy'
is used correctly for geometry and that a degeneracy
is a twist between two lines.
  It follows that for all knots that imply mathematical knots:
If its a prime knot presentation and it has no twists, then it is a Turks-head knot.
What is a "twist"?  E.g., does Rolfsen 4_1 have a twist?
(In normal parlance, I think it would be said to do so.)
But, then, 5_1 seems to be purely a twist?!

And, again, how do you match (close) a physical
/practical knot to get its corresponding match in
the Rolfsen Table?  You assert some matches above,
so how do you do this?


--dl*
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Dan_Lehman

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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. ... .
...

[X., I think that we should be indulging this discussion
in the other thread, as we aren't focused on the
OP theme any more, and getting too involved in
the other thread's theme here to deprive it of
proper material there!  --another copy & move.

--dl*
====]

Stagehand

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Thanks Dan,
You deserve this response right away.  Yes there are non- alternating prime knots in the Rolfsen knot table.  I do not  know what to do with non- alternating prime knots.  I carelessly assumed they arose at knots of 11 crossings but they come up beginning at knots of 8 crossings just where you found them listed after the polyhedron knots.  I make mistakes like this. Thanks again.