Author Topic: Non-TIB handcuff-type binders.  (Read 1948 times)

xarax

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Non-TIB handcuff-type binders.
« on: March 16, 2014, 09:35:36 AM »
  As we would need to attach the two returning eye legs to a nipping / constricting knot tied at the middle of the binding line, it is preferable if this knot is TIB. The knot at the middle of the binder shown at (1), which is based on two opposing crossing knots, is TIB. However, if we have a short line, or we can tie our knot near the one end of the line, we can also use non-TIB nipping / constricting structures. In this thread the reader can see two of the many possible similar binders based on a family of stoppers, composed of two interlinked, by two interconnecting bights, nipping loops. ( See the attached pictures ). The returning eye legs / Tail ends of the handcuff-type binders are nipped into those stoppers, each one by one nipping loop and one interconnecting bight.
   These stoppers can be tied in 8 different ways. There are two end crossing points, and two pairs of middle crossing points, corresponding to the crossings between each one nipping loop and each one interconnecting bight. So, taking account the over / under permutations, we have 2 x 2 x 2 x 2 = 16 different possible stoppers, but half of them being identical to the other half, due to symmetry, and half of them are non-symmetric. ( To be symmetric, the over / under relation at the last crossing point is determined by the corresponding relation at the first, so there are only 2 x 2 = 4 symmetric stoppers ). Also, the returning eye legs can penetrate the two openings, the opening of the one nipping loop and the opening of the one interconnecting bight, entering through any of their two sides, in 2 x 2 = 4 different ways. However, it turns out that only 3 of those ways form stable knots, so we can have "only" 8 x 3 = 24 different binders  :), 4 x 3 = 12 of which are symmetric. The one quarter of all the possible stoppers / binders ( 2 x 3 = 6) ARE even TIB, but, unfortunately, those which are TIB are also not symmetric - and we need symmetric stoppers / binders, because our arrangement is symmetric. With a not-symmetric nipping / constricting structure, the one returning eye leg / tail end would be nipped less than the other, so it will be the weak link of this two-link chain.

  In the pictures that follow I show two of them, that caught my eye. I have not tested each and every one of those 24 binders against each other to see which is more secure, of course, but I believe the differences will not be great. The returning eye legs are immobilized in between two opposing bights, the one formed by the corresponding nipping loop, and the one by the corresponding interconnecting bight, so they are immobilized very effectively. Considering the simplicity of the central nub and the easiness of its tying, I think that all those binders are secure to a satisfactory degree.

   1. http://igkt.net/sm/index.php?topic=4819.msg31391#msg31391
« Last Edit: March 17, 2014, 11:56:21 PM by xarax »
This is not a knot.

xarax

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Re: Non-TIB handcuff-type binders.
« Reply #1 on: March 16, 2014, 09:50:03 AM »
1.
This is not a knot.

xarax

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Re: Non-TIB handcuff-type binders.
« Reply #2 on: March 16, 2014, 09:51:15 AM »
   Same binder, tied on a stiffer, sleeker material.
This is not a knot.

xarax

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Re: Non-TIB handcuff-type binders.
« Reply #3 on: March 16, 2014, 09:58:07 AM »
  One of the other possible binders, based on a slightly different such stopper, the Bb2. ( I denote by A or B the "over" at the first and the "under" at the second middle crossing points - the one should be "over" and the other "under", or vice versa, so the nipping loop and the interconnecting bight are linked. Also, by a and b the "over" and "under" at the third and at the fourth crossing points. Finally, by 1 or 2 the "over" and "under" at the last, the fifth crossing point. I do not count the zeroth = sixth crossing point, because it generates symmetric = identical stoppers. The Ab1 and Ba1 are TIB .)
« Last Edit: March 17, 2014, 12:28:57 PM by xarax »
This is not a knot.

xarax

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Re: Non-TIB handcuff-type binders.
« Reply #4 on: March 17, 2014, 04:17:25 PM »
  Some pictures of another member of this family of binders, the Bb1 ( 1 = under, 2 = over, 3 = under, 4 = over, 5 = over ). It is topologically equivalent to the overhand knot - so, regarding the sought TIB, close but no cigar. Easy to tie and to inspect, due to its symmetry, and quite tight=secure on its returning rye legs / Tail Ends.
« Last Edit: March 17, 2014, 04:22:21 PM by xarax »
This is not a knot.

xarax

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Re: Non-TIB handcuff-type binders.
« Reply #5 on: March 17, 2014, 04:18:00 PM »
2.
« Last Edit: March 17, 2014, 04:22:48 PM by xarax »
This is not a knot.