Author Topic: 6_3 knot / "stoppper", and its less symmetric relatives  (Read 5935 times)

xarax

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6_3 knot / "stoppper", and its less symmetric relatives
« on: December 15, 2013, 09:59:31 PM »
 If we look at the tables of Mathematical knots, we see that, where a knot can be "represented" by a symmetric diagram, mathematicians prefer to show it this way - although mathematical knot theory studies the topological, and not the geometrical properties of the knots. However, I noticed that the third knot with six crossings, the 6_3, is always (?) shown in a asymmetric form ( the 6-1 and the 6_2 are always shown in a symmetric form ). On the contrary, with those same knots, the 6_1, 6_2 and 6_3, when viewed as we view them ( i.e., when we "cut" them somewhere, and we represent the two ends of this cut as the two ends of the "open" knot / "stopper" ), the situation is reversed ! The 6-1 and the 6_2 are less symmetric, and the most symmetric is the 6_3. ( See the attached pictures ). I do not understand why this happens, but I thought that it is an example of the different ways the same things can be viewed and represented, if they are examined by different people.
This is not a knot.

SS369

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Re: 6_3 knot / "stoppper", and its less symmetric relatives
« Reply #1 on: December 15, 2013, 10:24:24 PM »
It is amazing that any image viewed by multiple people can get interpreted so differently.

Looks like the making of a good loop knot collar (to me).   ;)

SS

Dan_Lehman

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Re: 6_3 knot / "stoppper", and its less symmetric relatives
« Reply #2 on: December 17, 2013, 11:43:55 PM »
...  I noticed that the third knot with six crossings, the 6_3, is always (?) shown in a asymmetric form ...

Is it because giving that extra twist of the center
parts will give a false (increased) crossing count?!

Working out topological equivalences is a major problem
--one that earned an IGKT member a Fields Medal (!)
for his contribution.  Of the trio of "Fig.9" forms that
I presented elsewhere and recently referred to
(where two are symmetric, and the 3rd is the form
that rockclimbers & cavers have dubbed "Fig.9"),
I've long known their equality yet have many times
been greatly frustrated in moving from one to another!!
--yes, in a case where I know that the conversion
is possible : so, finding things where one isn't so sure,
is just trebly troublesome.

Note that converting a practical knot into a topological
one is non-trivial : e.g., how to do so with the common
bowline?!  At least, IMO, one connects the two ends;
but it makes a difference as to which way one takes
the tail from between the eye legs --distinct knots
result!


--dl*
====

Luca

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Re: 6_3 knot / "stoppper", and its less symmetric relatives
« Reply #3 on: December 18, 2013, 03:30:58 AM »
Hi,

Starting from a Bowline with the tail coming out as shown by Alan Lee, we arrive at the 6_3 stopper!

http://igkt.net/sm/index.php?topic=4276.msg26587#msg26587

                                                                                                                            Bye!





xarax

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Re: 6_3 knot / "stoppper", and its less symmetric relatives
« Reply #4 on: January 06, 2014, 08:31:44 PM »
  Of the trio of "Fig.9" forms that I presented elsewhere and recently referred to (where two are symmetric, and the 3rd is the form that rock-climbers & cavers have dubbed "Fig.9"),

  Yet only one ( the third / "red" ) of the three Fig.9 forms, shown in the first post, is symmetric - when the knots are presented this way, i.e., as "stoppers" ( as "knotted" open strings in between two aligned and pointing to opposite directions tensioned ends ). The other two ( the "white" and the "yellow") can not even be considered as symmetric-to-each-other...
 
  A naive "explanation" is that the parent closed knot ( the 6_3 ) happens to be "almost" symmetric, and that the"cut" is sufficient to restore a complete symmetry, which was somehow hidden / broken within the closed form - but which now is free to manifest itself. On the contrary, when the parent closed knots are completely symmetric right from the start ( as it happens in the 6_1 and 6_2 ), a "cut" can not but destroy the existing perfect symmetry. So, when "cut" and "opened up ", the perfectly symmetric closed knots become asymmetric open "stoppers"- while the "almost" symmetric one is helped to reveal its symmetric nature, which was hidden / broken by the initial closure.
  We can test this "explanation" on the only non-symmetric of the 7 knots with 7 crossings - namely, the 7_6 ( see the attached picture ). When "cut" and presented as a "stopper", does it become symmetric, too - just like the 6_3 "stopper" ( the "red" one) ?
  At first I though that it does, but now I am not satisfied by the degree of the manifested symmetry any more- so I have to suppose that the previous naive "explanation", even if it seemed reasonable in the case of the knots with 6 crossings, it does not work at the next level, the knots with 7 crossings...
« Last Edit: January 06, 2014, 09:13:16 PM by xarax »
This is not a knot.

xarax

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Re: 6_3 knot / "stoppper", and its less symmetric relatives
« Reply #5 on: March 12, 2014, 11:30:41 AM »
   A useful reference, where all the fig.8-/ fig.9-like stoppers, up to knots with 9 crossings, by P-V Koselef :
  http://www.math.jussieu.fr/~koseleff/knots/kindex1.html
  http://www.math.jussieu.fr/~koseleff/

  See the reproduced from this page attached pictures of the so-called "Chebychef (tying) diagrams" of the fig.9 knots, which are very useful for the practical knot tyer, as they show the knot in a loose, yet "folded" form.

 
This is not a knot.