Author Topic: A measure of the simplicity or complexity of a knot.  (Read 7048 times)

xarax

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A measure of the simplicity or complexity of a knot.
« on: January 05, 2012, 02:29:58 PM »
  When we say that a certain knot is "simpler" than another, what exactly do we mean ? There are four different things that one can take into account, in order to measure this elusive "simplicity" ;
   1. Rope length. We can claim that a knot that consumes less material, is simpler. However, when  are we supposed to measure this rope length ?  Right after the initial set-and-dress phase, or when the knot is tightened to its strength limit?
   2. Number of tucks. Most knots are tied by driving the working end through loops, a number of times in a row. Knots that require a smaller number of tucks are often easier to tie, so we can say that they are simpler. However, it is easier to pass the working end through the same loop two or three times in a row, than through two or three different loops - unless those two different loops are one and the same two or three coiled "neck' ! So, this measure of knot s simplicity has its own twists, too.
   3a. Topology (genus) of the knot. A bend formed by two interlinked overhand knots can be considered simpler than one based on two double overhand knots, and/or two fig. 8 knots. However, one can argue that a double overhand knot is in fact simpler than a fig. 8 knot, although it is formed by twice the number of tucks...
   3b. Topology (crossing number) of the knot. In a lose, flattened knot, there is a certain number of points where the strands cross each other, go over or under each other. A small number of crossing points mean an easier to represent in 2D  knot...but the final form of the knot has often little relation with the initial form ( as it happens in the case of the Carrick bend, for example).
   3c. Topology (winding number). In a bend or a hitch, if we count the number of times the working end turns around the other link s strand or around the pole/line, we can have yet another measure of simplicity. A knot that utilizes friction effectively, without having too many turns and twists, should probably be thought as a simple knot.
   4. Total curvature of the rope s path. Who ordered this:) I think that this is perhaps the best way to measure the simplicity of a knot - because it is a measure of how much this knot is convoluted in 3D space, how many "turns" the working end has taken, to arrive at the tail, starting from the standing end.
   { We can think of the total curvature as a number that is the sum of all the angles formed alongside a curved path. To be more precise, for a space, 3D curved path - like the one followed by the rope inside the knot s nub - we first have to consider a straight line tangent to this path, and the 2D curved surface that is formed by this line, as it walks alongside a point that moves on the path (starting from the standing end and arriving at the tail). This surface is called "developable", because it is a "ruled" surface, it is "made" by straight lines the one next to the other - a curved surface that can be flattened on a 2D plane nevertheless. In short, a surface that can be made by a sheet of paper or metal, like a cylinder or a cone, but more freely formed. Now, if we flatten this surface and place it on a plane, we can measure its 2D total curvature, which is the total curvature of the initial un-developed space (3D) curve. }
   
« Last Edit: January 06, 2012, 07:24:17 AM by xarax »
This is not a knot.

Dan_Lehman

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Re: A measure of the simplicity or complexity of a knot.
« Reply #1 on: January 07, 2012, 07:42:32 AM »
And aside from launching Mike in Md. to venture into really
complex reading, what does any of this do for the knot tyer?

Thinking from a logical point of view , one might start
some classification based on the number of pieces of flexible
material comprised by the knot --e.g., a stopper has 1, but
eyeknots & end-2-end knots 2, and hitches 1 plus some object,
and so on.  (And maybe one has some notions of "first order"
& "second order" ... ?  Twin-eye knots will have 3.

!?

--dl*
====

xarax

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Re: A measure of the simplicity or complexity of a knot.
« Reply #2 on: January 07, 2012, 09:02:05 AM »
...what does any of this do for the knot tyer?
...one might start some classification

  Iff you are a knot tyer, you already know !   :) Unless you prefer to have all those hundreds of practical knots thrown  into a large, dirty bag, and only labelled by an ABoK number, or a silly given name...Classification, the first thing any science should do, as you answer by yourself, right after the rhetoric introduction. But not only this... We should be able to compare apples to apples, and similar simple knots with similar simple knots... so the question posed by a knot4u s remark a few days ago, really arises immediately. What exactly is a "simple" knot, i.e, is something out there - or perhaps more than one thing - that can be really measured, objectively, scientifically, and define "simplicity" of knots ?   
   I plan to start a comparative test of bends ( and possibly of bowline-like loops, too) strength, by destructive tests, using a sort of a universal material testing machine I am building these days. I was asking myself "what exactly are the things you are going to measure on those knots, beside their breaking strength ? Their rope length ? The number of turns of the working end around itself, or around the other link s working end ?
   I believe anybody understands that we should classify our knots in a way more useful and descriptive than ABoK numbers and names given by historic accidents, and this thread is meant to address this issue - and the problems we would probably be confronted with during any such attempt.
« Last Edit: January 08, 2012, 07:44:24 PM by xarax »
This is not a knot.

xarax

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The minimum-total-curvature conjecture for maximally tightened knots
« Reply #3 on: January 09, 2012, 12:42:14 PM »
  Regarding this measure of knot simplicity I propose ( total curvature of the rope s path inside the knot s nub), I have another thing to add, which might be of some interest to somebody.
   A tight knot is nothing but the knot where the rope length is minimized. So, if we have the same two knots ( tied on ropes of the same diameters, of course ), the one being more loose than the other, we can actually measure  this fact : we have just to measure the relative rope lengths of the two knots, and compare our readings. The tighter knot will be the knot with the shorter rope length. Ideally, if there was a way to know the exact minimum rope length such a knot can have, we could, in principle, know if a knot is fully tightened or knot. In theory there is such a minimum, (otherwise the knot should have been able to straightened to a straight line !), but, unfortunately, it has not been calculated yet for any knot - not even for the simpler ones.
   Intuitively, I have been driven to the conclusion that, if a knot is maximally tightened ( so its rope length is minimized ), its total curvature should be minimized, too. I dare to say this because, when we tighten a knot pulling its ends, we do not only trim the excessive material used (i.e. we minimize the rope length ), but also we stretch the material as much as possible ( i.e. we minimize the total curvature). This was another factor that persuaded me to propose the total curvature as a most general measure for rope s simplicity.
   Of course, as I am no mathematician, I by no means can prove or disprove  this conjecture... I just hope that somebody, someday, will learn about it and would enlighten us here.
 
« Last Edit: January 09, 2012, 12:46:26 PM by xarax »
This is not a knot.

SS369

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Re: A measure of the simplicity or complexity of a knot.
« Reply #4 on: January 09, 2012, 04:35:26 PM »
Of course to bring the knot to the point of minimal size, one will have to take the subject knot to the nanosecond before the big bang of destruction. At the moment just before this destruction point (the smallest it can be), has the knot been detrimentally weakened?

Which begs the question as to what will be the "standard" test medium.
What cord or rope, of what material and construction will be used?

Sweeney

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Re: A measure of the simplicity or complexity of a knot.
« Reply #5 on: January 09, 2012, 07:37:15 PM »
I am not sure it matters what rope is used unless there is an attempt to derive a standard against which all knots are measured nor is it necessary to load the knot to breaking point. To test knots A, B and C for example one would use the same rope for all 3 and load each one with a specific force sufficient to tighten the knot but well within the BS of the rope. This gives a comparison of rope consumption for the 4 tested knots I think one can assume that the consumption of a different cord will show that all 4 knots keep their relative postions (in doing this it is important I think to avoid unduly ealstic cord which may compress and stretch excessively under load eg bungee cord).

Having established the relative (but not absolute) rope consumption of A, B and C one could use these 3 as comparison knots for any other knot and therefore establish a table of relative rope consumption. All that is required is the same rope to be used for all knots in a given test (only one of A,B and C may suffice but using all 3 must be safer until some consistency is observed).

To try and establish the absolute minimum knot size may be possible by tightening the knot with a given force then measuring the diminishing level of rope consumption as more force is applied. One would think that the knot would tighten to close to its minimal size and then the amount by which the rope is no longer consumed would diminish rapidly until the rope broke. Having established the mean breaking point one could take the force-x (where x is a unit of force sufficient to get close to breaking point but enough to avoid that) and project minimal size mathematically (note - I didn't say how!).

The obvious problems are standardisation of rope type and rope elasticity - the rope consumption could seem to be negative if the rope elongated enough under strain. This is less of a problem with relative measurement as one would assume the same degree of stretch outside the knot applying to all knots tested. Although this may be possible experimentally adding knots to the list would be well nigh impossible as the test conditions could probably not be replicated.

Just a thought.

Barry

xarax

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Re: A measure of the simplicity or complexity of a knot.
« Reply #6 on: January 09, 2012, 07:39:35 PM »
At the moment just before this destruction point (the smallest it can be), has the knot been detrimentally weakened?

  Nature has been kind with us - just because "natura non facit saltus". There is always a gradual transformation of the one thing to the other, from the one state of things to the other state of things. (*) So, there is always a - however thin - time "skin", that protects the existence from the destruction.
   Speaking about the tensile properties of the materials used for ropes - that are not brittle -,  there is a relatively long time period where the material behaves like a spring - its elongation is directly proportional to the applied force -, then there is a very short time period until it reaches the point of "yield strength", then a longer one until it reaches the point of "ultimate strength", then a period of gradual deterioration, and then, and only then, comes the abrupt deterioration, the fracture, and the final destruction. (2) So, between the point of the maximum yield strength and that of the fracture, there is plenty of time to tie another knot on another rope !   :)
   More specifically, I am persuaded by Tom Moyer s article that the most versatile and forgiving material, is no other than the old, cheap nylon ! (4)  If we wish to get rid of the side-effects of the complex double braided ropes ("milking", for example), the single/solid braided ropes is the next more natural choice. They keep a satisfactory round cross section, they do not present the peculiar behavior - due to their handness- of the laid ropes, and they behave like one unified element - so they are more representative of the archetypal general purpose rope, I believe.
   
1.  http://en.wikipedia.org/wiki/Natura_non_facit_saltus
2.  http://en.wikipedia.org/wiki/Ultimate_tensile_strength
3.  http://user.xmission.com/~tmoyer/testing/Qualifying_a_Rescue_Rope.pdf
4.  http://user.xmission.com/~tmoyer/testing/High_Strength_Cord.pdf

* At least at the classical/non-quantum world
This is not a knot.

xarax

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Re: A measure of the simplicity or complexity of a knot.
« Reply #7 on: January 09, 2012, 08:02:41 PM »
   Thank you Sweeney,

...establish a table of relative rope consumption [for each knot]

...at some specific proportion(s) of maximum rope strength, 1/4, 1/2 and 3/4, for example.  Very good idea... and a lot cheaper than mine, too !  :) Especially if one uses nylon ropes - which seems to retain their strength, respectably of the flex cycles. I would think about it, make some tests, and see how it goes.
 
measuring the diminishing level of rope consumption as more force is applied.

  Another very good idea...It sounds interesting and reasonable to me, because the difference in the behavior of the material at the area underneath the elastic and proportional limit, and over the yield strength limit, should be easily noticed.   


This is not a knot.

Dan_Lehman

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Re: The minimum-total-curvature conjecture for maximally tightened knots
« Reply #8 on: January 10, 2012, 07:18:06 PM »
A tight knot is nothing but the knot where the rope length is minimized.
 ...
 Intuitively, I have been driven to the conclusion that, if a knot is maximally tightened ( so its rope length is minimized ), its total curvature should be minimized, too. ...
   Of course, as I am no mathematician, I by no means can prove or disprove  this conjecture... I just hope that somebody, someday, will learn about it and would enlighten us here.

Consider the case --e.g., "Fig.9" (the almost-a-stevedore)--
where a topological entity can assume various orientations?
Each of these can get "maximally tight" without changing
into each other.  One can rearrange the *double* turns of
the double bowline in interesting variety, too --creating
a helix or a sort of spiral vice the adjacent turns, all of which
can be made tight.
!?

The hope that effective classification will simplify some other
chore crashes into the so-far unwelcoming rocky shoreline
that shows the former to be anything but simple, itself!


--dl*
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xarax

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Re: The minimum-total-curvature conjecture for maximally tightened knots
« Reply #9 on: January 10, 2012, 10:08:33 PM »
a topological entity can assume various orientations?

= dressings. Yes, of course ! A knot can have a number of dressings, in all of which it can be maximally tight, although it does not consume the same amount of material - its rope length is different. However, I guess that this number will always be a natural number, i.e. it will not be infinite, and also that it would be a very small  number. We can say that we have a mechanism that settles in more than one states of equilibrium, but the amount of the energy consumed in each state is not the same. However, to pass from a higher energy state to a lower one would not be possible, if those states are separated by an energy barrier higher than both of them !  So I guess that we should have to classify these topologically identical dressings of knots, as different knots... but we should really bother about it, because we are speaking about our Nodology/Desmology here, not their knot theory !  :)

The hope that effective classification will simplify some [things, is itself] anything but simple !

   Whoever told you that speaking about simple things, in general, and simplicity itself, in particular, would be a simple thing, was a very perplexed man !  :)

   I repeat my proposal in a few words : In each knot /dressing of a knot, there is a specific quantity that can be measured objectively and precisely, the total curvature. It is a measure of how much a knot is convoluted in space, and I believe that it is the best single measure of the simplicity of a knot/dressing of a knot. 
« Last Edit: January 10, 2012, 10:11:54 PM by xarax »
This is not a knot.