The Span bend and crossing-knot loop(s).

[xarax]

   In the Englishman knot(s), we can replace the two overhand knots (ABoK#1038), or the two fig.8 knots (ABoK#1040), with two interlinked crossing-knots. ( It was the geometry of the initial overhand knots or fig.8 knots we needed - so the ends of the loops could be locked securely inside the nub(s) by following convoluted enough paths - not the topology ! ) Doing this, we manage to transform the Englishman knot(s) into the PET TIB loop(s) shown in this thread.
   We can see that the four limbs/ends of those two eyeknots stem out of the same nub. So, we can now consider this nub as the "original" bend, and the two eyeknots as transformations of this bend into loops. ( Notice that this bend is asymmetric : so, if we connect the one pair of the Standing end of the one link and the Tail end of the other link, we get the one loop, and if we connect the other pair, we get the other loop ).
    Why do I call this bend " Span bend " ? Simply because the lines of two crossing-knot loop can be rearranged within their nubs, and generate : 1., the well known, easily tied and untied Span loop (ABoK#1049), which I call Span A -  and, 2., the less well known but also easily tied and untied ABoK#1055 loop, which, due to the ( common with the Span loop ) way it is generated from the "original" Span bend, I call Span B (*). I have not seen either the Span loop or the ABoK#1055 having been transfigured / transformed into those crossing knot PET TIB loops.

(*) I had not noticed that the loop I used to call "Span B" was, in fact, the ABoK#1055, till it was pointed out to me by Luca. Thank you, Luca.

P.S. If we interchange the Standing and the Tail Ends, this loops becomes one of the Samisen TIB bowlines :
       http://igkt.net/sm/index.php?topic=4883.0
       See also :
       http://igkt.net/sm/index.php?topic=4890.0

[xarax]

  The Span crossing-knot loop (A).

  P.S. The inverse of this loop, is the same Samisen TIB bowline :
         http://igkt.net/sm/index.php?topic=4883.0
         See also :
         http://igkt.net/sm/index.php?topic=4890.0

  P.S.(2) By dressing it differently, we can arrive at the reversed Right Angles loop, shown at (1), or at a variation od the 1 loop, shown at (2).
  1. http://igkt.net/sm/index.php?topic=4422.msg32243#msg32243
  2. http://igkt.net/sm/index.php?topic=4603.msg29705#msg29705

[Luca]

Hi xarax,

I noticed that the Span crossing knot loop B is topologically the same of ABoK #1055 ( basically the "false collar" around the two legs of the eye of #1055 is "backflipped" to get the loop you show us).So, since I consider the Farmer's loop (#1054/56) as a loop "brother" of # 1055 (there is a similar difference between the Tugboat A and Tugboat B regarding the orientation of the nipping component compared with the collar component),I think it might be interesting to try to run a "backflipping" of the "false collar" around the two legs of the eye of #1054/56,and then try to derive the "corresponding" asymmetric bend and the other loop "corresponding" to the Span loop.

                                                                                                                Bye!

[xarax]

...the Span crossing knot loop B is topologically the same of ABoK #1055

...because each and every TIB knot is topologically the same as any other TIB knot !    0 = 0, always...
   If you enjoy the non-obvious correlations, ( as I do  ),  notice that the ABoK#1050 is the "inverse" of the ABoK#1051, in the same sense as the Span crossing knot loop (A) and the Span (A) loop ( ABoK#1049) are the "inverse" of the Span crossing-knot (B) and the Span (B) - ABoK#1055 loop. The ABoK#1051 is a simplified, yet "unexpectedly" TIB Eskimo-like loop, which deserves further study. I had tried, but I have not managed to secure it in a decent yet simple way...
   Now you have mastered the "backflipping" - as you call the trick by which we reeve the whole nub through a last bight, which "locks" the TIB knot and becomes a collar around the pair of the eye legs ( a "low' collar ), or around the pair of the Standing end and Tail end ( a "high" collar ) -, you can apply it to many more cases !    ( See the thread about the "Sheepshank loop" ( with a "low" collar ), and the "Sheepshank bowline" ( with two collars, a "low" and a "high" collar), where I have used the same trick.
   Frankly, the TIB eyeknots with a "low" collar ( another one that comes to mind is the Jug Sling / Coming and going TIB bowline(s) ) are not very interesting as general fixed loops,because the angle between the two eyelegs has to be small - and, of course, they are not meant to be ring-loaded at all. To my view, if we need something more secure than the pet TIB loop, we better use a different type of secure TIB bowline, without a "low" collar.

[Luca]

Hi xarax,

...the Span crossing knot loop B is topologically the same of ABoK #1055

...because each and every TIB knot is topologically the same as any other TIB knot !    0 = 0, always...

You're right, this is true, and I should explain myself better, because all the TIB knots are topologically equivalent to the unknot, but it is also true that,to transform a TIB knot in another, on the one hand there are cases in which a "simple" motion such as the "backflipping" (it is actually a term by Dan Lehman that maybe sometimes I'm using inappropriately) of a collar may be sufficient,while the other extreme, there are cases in which it will necessarily have to reach a "true" unknot and realize the knot from the start! (or at least I think ... I'm not 100% sure!)
In my opinion #1055 is at the same level of complexity of  #1056. I attach a diagram of the bend that I obtained starting from # 1056, as I explained (badly) in my previous post (which, as bend is weak, but that maybe can be interesting in order to derive the relative eye-knots): 

                                                                                                                                Bye!

[xarax]

   I understood what you meant, but I was just kidding...   There is the topological complexity ( unknot, overhand knot, fig.8 knot, fig.9 knot, and so on...) and the geometrical complexity - which has to do, in my opinion, to the total number of turns of the lines in 3D space. ( Read (1)). Imagine you are a creature walking along the line, and measuring the sum of the angles, relatively to the fixed stars, of all the turns you make to go from the Standing end to the Tail. If you make many and/or sharp turns, this sum would be large, so you will know that you were walking along a line of a geometrically complex, convoluted knot - or that you had consumed too many drinks last night ! 
   Now, if you measure this number, you can really tell if a knot is more complex than another, or not. Mind you we are not talking about the simplicity/easiness or complexity/dificulty of the tying of the knot here, but for the simplicity or complexity of the (geometric) shape of the knot, i.e, of the knot per se  .
   
  Ooops ! You are right, and I thank you very much for this : I had not noticed that the Span (B) loop is the ABoK#1055 ( which is less well known than the Span (A) loop ) - and I had not even figured out the clever way to tie it, starting from a shape 8 unknot and reeving the bight trough it, shown by Ashley. I was tying it by the same "three bights" method by which I tie the Span and the pet loop. I have not made up my mind yet, about which loop I prefer, the Span (A)(ABoK#1049) or the Span (B)(ABoK#1055), but I think that the pet loop cow is better of both !   

1. http://igkt.net/sm/index.php?topic=3740

[xarax]

...a diagram of the bend that I obtained starting from # 1056...which, as bend is weak, but that maybe can be interesting in order to derive the relative eye-knots): 

   The weak part is the blue Tail... If you just translate it "over" the red Standing part, to the other side of the collar of the red crossing knot link, you will see that the bend becomes much stronger. See also the ABoK#1451 loop, at (1). ( And if you do not tuck it through the red collar at all, but through the central opening - where it is nipped by the two opposing bights most effectively - the bend becomes rock solid  , but the corresponding crossing knot loops are not TIB any more...  )

http://igkt.net/sm/index.php?topic=4013

[Luca]

Thanks for that!

One thing that at this point I can think of,is that I personally would be more inclined to consider as an alternative version of the Span crossing knot loop (A) the loop above,rather than the "inverse" version (B) at your first post,that precisely because of this "inversion" eye/ends,from my point of view turns out to be a knot with different characteristics(I am reminded of the example of the relationship between the DL locked Bowline and the "inverse" Perfection loop).
Consequently, I would rate the Span crossing knot loop (B) and the unreliable loop  that you mention above in a slightly different family of loops,as indeed personally I would put #1055 more directly related to  #1056 rather than #1049.

                                                                                                                    Bye!

P.S.It is only right for me to remember that, if I came easy to notice certain relationships with abok # 1055, actually was thanks to mat http://igkt.net/sm/index.php?topic=1870.msg27094#msg27094 ,TMCD http://igkt.net/sm/index.php?topic=4333.msg27133#msg27133 ,and Ruby http://igkt.net/sm/index.php?topic=4333.msg27243#msg27243 .
   

[xarax]

this "inversion" eye/ends... turns out to be a knot with different characteristics

  True - but this happens with all "inversions" !    See it as a strictly technical term, describing only the way the two eyeknots are generated from the same bend, not the eyeknot themselves. Use the same asymmetric bend, and form an eyeknot  by connecting the one or the other pair of ends. The "inversion" denotes this "or", nothing more. It does not mean anything for the characteristics of the individual knots.
   The meaning of this thread is to present the bend, the two midline loops ( ABoK#1049 and ABoK#1055) and the corresponding two crossing knot TIB loops - so, five, in total, knots - in one coherent scheme. I always try to find knots that do not come out of the blue, but appear in a "natural" way, if we just explore systematically the consequences of a simple knotting idea. The idea was that there are midline/TIB loops that can be "back-flipped", and be transformed into crossing knot end of line TIB loops. Moreover, that there are pairs of such midline/TIB loops. which are based on the same asymetric bend, the one being the "inversed" of the other.

[Luca]

My intention was not at all to break or obscure in any way the coherence of this scheme, but only to express the point of view that conceptually (OK, I know:"conceptually" is a big word for someone like me! ), to call "Span crossing loop knot A" (OK) and "Span crossing knot B", the loops that you show us above, to my ears it sounds a bit like to call "Locked bowline A"  the DL locked Bowline (OK), and "Locked bowline B "the Perfection loop.Only this basically I wanted to notice!

[xarax]

   I see what you mean - but the common "original" bend at the heart of the flipped version of the Perfection loop and the flipped version of the Lehman s TIB locked bowline can not serve as a basis of any other secure loop - not even one, I am afraid. That means we can not have two secure loops, to complete the set of four loops generated by the same "original" bend ( so, related to each other ), as it happens in the case of the four loops we are talking about in this thread.
   You point out that if you "front-flip" the Lehman s TIB bowline and you "back-flip" the Perfection loop, you arrive at two new loops, the one the "inversed" of the other, that shear the same nub. However, this nub can not serve as a basis either of a secure bowline, or of a secure crossing-knot loop. Considered as a bend, it is composed of one crossing-knot link and one nipping loop link. Its weak link / point is the nipping loop : any combination of two loaded ends from the one side and one loaded end from the other side you choose to form an eyeknot, the nipping loop has the tendency either to slip through the crossing-knot, or to degenerate into an open helix too easily.