Author Topic: Bends Based on the Carrick Bend Diagram  (Read 151 times)

NautiKnots

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Bends Based on the Carrick Bend Diagram
« on: July 14, 2017, 11:07:55 PM »
A couple of years ago, I was perusing my copy of Ashley's Book of Knots and came across the following passages:

"Many bends in common use may be tied on the Carrick Bend diagram" (ABoK 1545).  As examples, Ashley gave diagrams for the Reef Knot (1549), Sheet Bend (1550), Carrick Bend (1551), and Granny Knot (1552).

"In a knot of eight crossings, which is about the average-size knot, there are 256 different 'over-and-under' arrangements possible...  Make only one change in this 'over-and-under' sequence and either an entirely different knot is made or no knot at all may result".  Ashley gave illustrations for the Reef Knot (77), Sheet Bend (78), and no knot (79).

That got me to wondering just how many bends there are in the Carrick form, and were any of them new.  Disregarding the lead (as in the Josephine Knot) the Carrick diagram has eight crossings, and as Ashley observed, there are potentially 256 over/under combinations (see the first image below).  Add to that, however, that (as a bend), the standing ends may be either on diagonally opposite sides (see image 2) or on the same side (see image 3), and you get a total of 512 possible bends (1024 if you entertain all possible lead combinations, but rotational symmetry allows us to discard half of them).

Being both curious and methodical, I decided to tie them all and find out.


NautiKnots

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Enumerating the Knots
« Reply #1 on: July 14, 2017, 11:31:54 PM »
Then the question arose - how to enumerate the knots?  That is, what method could I use to be sure I tied all of them without duplication?  I came up with the diagrams below.  The first image shows a naming scheme where I assigned a letter (A-H) for bends with opposite leads.  The second is the same for bends with leads on the same side.  For the sake of convenience, the crossings are alphabetical in order of tying sequence.  That is, I would first lay out one working end with crossing "A", and then would use the other working end to make crossings "B" through "H".

Now, consider the string "LABCDEFGH", where "L" stands for "lead", and "A" through "H" stand for the crossings.  Assign the numeral "0" for diagonal lead, and "1" for lead on the same side.  For each crossing, assign the numeral "0" for passing under, and "1" for passing over.  That yields a 9-digit binary number.  (The first image below corresponds to possibility 000000000 (decimal 0) as the lead is opposite and all crossings go under.  The second image corresponds to number 100000000 (decimal 256).  The lead is the same and all crossings go under.)

Proceed sequentially from 000000000 (0 decimal) to 111111111 (511 decimal), and you'll enumerate all possible bends conforming to the Carrick diagram.

That's exactly what I did.  I cut a bunch of short pieces of black and white cord and tied each knot by number.  Most of them didn't form a knot.  That is, the resulting structure (like the two diagrams) were not self-binding.  I kept the ones that did form a knot and labeled them for later study.

It turned out that only 100 of the 512 permutations formed knots.  The others simply fell apart.

NautiKnots

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Identifying the Knots
« Reply #2 on: July 14, 2017, 11:54:41 PM »
Then I had to identify the knots.  Some were obvious, and others were unfamiliar.  On some, all the crossings made a difference and on others (such as the Reef Knot), some crossings did not matter.  One had crossings that could be twisted, and in one orientation might cause the knot to bind well but not in the other.  Some knots consisted of two right-hand loops or two left-hand loops.  Others had one of each.  Several formed bights (like the Reef Knot again) instead of loops (but I stuck to a convention of calling clockwise bights left-handed and counterclockwise bights right-handed).

In the end, I identified 31 (or 33 depending on how you count the Whatknot) unique knots in 7 basic forms.  The basic forms are all identified in Ashley's Book of Knots -- the 31 (or 33) knots are variations with different chirality (mirror images) or are tied with leads reversed (or both) from Ashley's diagrams.  I didn't discover any "new" bends, which really isn't surprising.

In subsequent posts, I'll attach diagrams showing the variations of each form, labeled by ABoK number and two possible suffixes.  "R" stands for leads reversed -- that is, if Ashley showed the standing ends on the same side, this knot has them opposite (or vice versa).  "M" stands for mirror image -- Ashley was not consistent in showing left- or right-handed versions of knots in his diagrams so sometimes "m" will mean right-handed and sometimes it will mean left - whatever is the mirror image of Ashley's drawing.
« Last Edit: July 15, 2017, 12:57:42 AM by NautiKnots »

NautiKnots

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Form 1: The Reef Knot
« Reply #3 on: July 15, 2017, 12:26:50 AM »
The most basic form is the Square or Reef Knot.  The first image below shows the reef knot as drawn by Ashley as bend #1402 (on the left) and its mirror image (1402-m) on the right.  Note the black diamonds.  Those indicate crossings that don't matter.  They are outside the binding area of the knot and will uncross naturally into bights instead of loops.

The second image is the Reef Knot with opposite lead, which goes by the name of the Thief Knot, ABoK 1207.  What's interesting to note is that the Thief Knot does not have left- and right-handed versions.  If you tie a mirror-image of the Thief Knot, and flip it end-for-end, you wind up with the original knot.   That is very unusual.  I doubt that the Thief Knot is unique this way, but it is the only knot I've noticed with this form of symmetry.  Of the 100 crossing permutations that formed knots, more of them made the Thief Knot than any other.

NautiKnots

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Form 2: The Granny Knot
« Reply #4 on: July 15, 2017, 01:24:41 AM »
The next general form is the Granny Knot.

The first diagram below shows the Granny Knot as illustrated as ABoK 1405 and its mirror image.

The second diagram is of the Whatknot - ABoK 1406 and 1407.  The left image is the knot Ashley drew -- the right one is its mirror image. The black diamond occupies a crossing that (depending on how you think of it) may or may not matter.  Ashley refers to both variations as the Whatknot, but notes that one crossing (1406) is fairly secure but the other (1407) is quite insecure.   Given that the knot (even when drawn up) may spontaneously morph from one form to the other, and that both geometries go by the same name, I'm inclined to call this 2 variations (1406/7 and 1406/7m yielding the 31 total)  If you prefer to call it four (1406, 1406m, 1407, and 1407m for a total of 33), that's fine too.

NautiKnots

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Form 3: The Sheet Bend
« Reply #5 on: July 15, 2017, 01:53:46 AM »
The Sheet Bend is a little unusual.  It is asymmetric end-for-end so there are two different ways to reverse the lead.  That yields the 8 variations shown in the two images below.  As with the Square Knot, the black diamond represents a crossing that does not matter (the bight).

Ashley only drew two versions of the Sheet Bend, ABoK 1431 and ABoK 1432 -- the so-called "Left-Hand Sheet Bend".  This is an unfortunate appellation, because each knot has both left- and right-handed versions.  ABoK 1432 is really 1431 with one of the leads (the bight) reversed, not opposite chirality.

The first image below shows 4 variations of the Sheet Bend, ABoK 1431, 1431-m, 1431-r, and 1431-rm.  The second image shows the same 4 variations of the False (a.k.a. "Left-Hand") Sheet Bend, ABoK 1432, 1432-m, 1432-r, and 1432-rm. 

To borrow terminology from the Bowline, you could call the reverse-lead versions of these knots "Eskimo" Sheet Bends.  I don't think Ashley drew these.

NautiKnots

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Forms 4, 5, and 6: Single Carrick Bends
« Reply #6 on: July 15, 2017, 02:58:28 PM »
There are 3 general forms of the so-called "Single" Carrick Bend.  These knots, although they superficially resemble the Carrick Bend, do not have a regular over-one-under-one pattern.  As they are all symmetric end-to-end, there are 4 variations of each.

The diagrams below show the three bends ABoK 1443, 1444, and 1445, each in the form (left to right) that Ashley drew it, mirror image, with the lead reversed, and a mirror image of the lead reversed.

Since all these bends are inferior to the real Carrick Bend, I really don't have much to say about them, other than if anybody asks "how many Single-Carrick-Bends are there", the answer is 12 (or 6, or 3 -- depending on how you count them).

NautiKnots

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Form 7: The Carrick Bend
« Reply #7 on: July 15, 2017, 03:30:18 PM »
Finally, we reach the last general form, which is the true (sometimes called "Double") Carrick Bend.  This is ABoK 1439, which Ashley called "perhaps the nearest thing we have to a perfect bend".  It also has 4 variations:  The version Ashely drew, it's mirror image, one with a lead reversed, and a mirror image of the one with a lead reversed (see the diagram below).

Ashley occasionally noted the difference between knots tied right-handed vs. left-handed, and predominately (but not consistently) drew right-handed versions.  His diagram of ABoK 1439 is actually the left-handed variant (although he drew the right-handed knot elsewhere).

The Carrick Bend with lead reversed (shown right-handed as ABoK 1428) is also known as the "Double Coin Knot" (when the standing ends form a loop), the "Pretzel Knot" (ABoK 2283); and also is the start to the Prolong Knot (ABoK 2242). the 3-lead 4-bight Turk's Head (ABoK 2287), and others.

The Carrick Bend with opposite lead serves as the start to a number of knots as well.  Those include the Chinese Button Knot (ABoK 600), the Knife Lanyard Knot (ABoK 787), and the 4-lead 3-bight Turk's Head. 

Together the two variations form the basis of a whole series of knots and mats known as Carrick Mats, Chinese Mats, and Basket-Weave Mats.  The Carrick Bend form keeps popping up, even in newly proposed knots, and is an incredibly useful knot to know.

That concludes my enumeration of the bends tied on the Carrick diagram.  I hope you found it interesting.

Regards,
Eric

knotsaver

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Re: Bends Based on the Carrick Bend Diagram
« Reply #8 on: July 17, 2017, 09:35:01 AM »
Hi Eric,
surely interesting.
(I've never had the patience for tying all of them)...
You should mention the wonderful Tumbling Thief Knot...


(http://www.surreyknots.org.uk/65-tumbling-thief-knot.htm

http://igkt.net/sm/index.php?topic=3716.msg27342#msg27342

http://igkt.net/sm/index.php?topic=3716.0)

Quote

To borrow terminology from the Bowline, you could call the reverse-lead versions of these knots "Eskimo" Sheet Bends.  I don't think Ashley drew these.

Those are usually called Lapp Bends
https://igkt.net/sm/index.php?topic=1955.msg13630#msg13630

Ciao and thanks,
s.
« Last Edit: July 17, 2017, 09:41:29 AM by knotsaver »

NautiKnots

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Re: Bends Based on the Carrick Bend Diagram
« Reply #9 on: July 18, 2017, 12:50:47 AM »
Thanks,

Do people generally consider the different forms in which a Whatknot or Thief Knot may be dressed to be different knots?  Each of them has a secure form and an insecure form (and nothing preventing it from morphing from one to the other).  Ashley illustrated two different arrangements (ABoK 1406/1407) of the Whatknot, but doesn't give them different names.

I didn't think of them as different knots, so I didn't note the properties of the variations.  If they are considered different knots, then I may need to retie the 8 Whatknots and the 12 Thief Knots  to see what they look like.


You should mention the wonderful Tumbling Thief Knot...
(http://www.surreyknots.org.uk/65-tumbling-thief-knot.htm
Thanks for the reference.  Am I mistaken, or is diagram 4(a) a Sheet Bend, not a Thief Knot?

Quote

Those are usually called Lapp Bends
I wasn't aware of that appellation.  Thanks again.

Is there something about the Arctic that causes us to name reverse-lead versions of the Bowline and Sheet Bend after Eskimos and Laplanders? lol.

Regards,
Eric

knotsaver

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Re: Bends Based on the Carrick Bend Diagram
« Reply #10 on: July 18, 2017, 10:46:49 AM »
Do people generally consider the different forms in which a Whatknot or Thief Knot may be dressed to be different knots? 

People should... The "topology" of the knots is the same, but the geometry is different.

Quote
Each of them has a secure form and an insecure form (and nothing preventing it from morphing from one to the other).  Ashley illustrated two different arrangements (ABoK 1406/1407) of the Whatknot, but doesn't give them different names.

But he gave different numbers!  ;)

Quote
Am I mistaken, or is diagram 4(a) a Sheet Bend, not a Thief Knot?


No, you are right! the diagram is wrong!? and the same wrong figure is on the index page!?

http://www.surreyknots.org.uk/igkt-knot-charts.htm

You are welcome.
Ciao,
s.

NautiKnots

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Re: Bends Based on the Carrick Bend Diagram
« Reply #11 on: July 20, 2017, 08:25:27 PM »
If one considers only topology, then there are just 2 Whatknots - one tied right-handed, and one tied left-handed (the mirror-image of the first).  If, however, one considers knot geometry (the shape of the knot when dressed), then there are several different Whatknots.  On the Carrick diagram, there are four ways that the end crossings may be arranged, each of which draws up to a different geometry, and each of which has a mirror-image.  That yields 8 geometrically distinct knots.

To add to that, there are arrangements of the Whatknot that don't conform to the Carrick Diagram (such as ABoK 1208 and ABoK 1459), each of which has left- and right-hand versions.  That makes 12 different knots, and it doesn't even consider capsized forms (the Clove Hitch).

Unless seized, as in the Reeving Line Bend (ABoK 1459) or tied in non-rolling material like the Grass Bend (ABoK 1490), the Whatknot can easily change from one geometry to another.  Even though some forms (ABoK 1406) are quite secure, others (such as #1406 pulled by the running ends) slip easily under very low load.  This makes the Whatknot a very untrustworthy bend.

There are 12 ways to tie the Thief Knot on the Carrick diagram, and I'll re-tie those when I get a chance.  Like the Whatknot, there are secure and insecure geometries.  I think some of these geometries also have left- and right-handed versions (unlike ABoK 1207) so that exercise should be interesting.

Regards,
Eric