Hi Patrick,
So you are proposing a classification which identifies that a knot comprises the same structural component over and again. OK, but to call a knot such as the bowline a "- disorder knot" is confusing, because it contains a structure even though it contains it only once.
Perhaps a more apparent classification would be to call a knot which has a structure only once an - Order(1) knot while a knot such as a three strand MWK would be an Order(3) knot and an N bight TH would be an Order(N) knot.
But wait, the Fishermans knot has two incidences of the strangle - is this an Order(2) knot?
And hold on - What about knots which have some structures only once while having other structures multiple times - the Sliding Grip Hitch has three or four round turns, so while part of the knot is Order(1), other parts are Order(3).
Naturally, this leads on to asking how small a structure you are proposing be considered as a repeating component - ultimately, all knots are made up of numerous small bent sections and numerous straight sections arranged in a variety of ways.
But has this classification got us anywhere?
Has it told us anything about a knot?
Yes, I am being critical of your suggestion, but this is because the area of classification is I believe totally ignored by knotting experts, while classification of structure and identification of functionality are what I consider to be the two most critical areas of knotting science yet to be addressed.
Without question, this omission is down to the fact that these are HARD problems. Math has addressed the problem and created a functional classification for knots. But that system is less than useless to describe the aspects of a knot relevant to their functionality as Force Vector Machines in the real world.
The Myrtle is such a simple knot - until you start to try to describe its functional structure. Your proposed classification would call it an - order knot, or an Order(2) knot... The Overs Index struggles but at a push might manage a classification of OI 9:14. Even its Binary signature 101000101//010111010, although capable of effectively pigeon holing the knot, fails dismally to describe the knots structure or operational functionality in even the slightest manner.
I believe that you are right to propose a classification which reflect component replication within a knot - but I also believe that we should be attempting to describe what is being replicated and how they are interacting.
The Myrtle is a good (i.e. simple) example to start with - it is just two interlocking round turns, so it might be described as RT(2). But this doesn't describe how they are interlocked. How would we do that? How would you describe where the two cords meet on the Round Turn? Perhaps we can call this crossing point the Origin., so we could state that the origin of the Round Turn was contained in the Wrap of the second Round Turn, while the origin of the second RT was contained within the Wrap of the first RT. How would we depict this intimate mating?
Do we need a shorthand for structures and a nomenclature for containment? RT = Round Turn; RTO = Round Turn Origin (the crossing point of the leads); B = bight (or half turn) etc.. But what about containment? How would we depict that the RTO was contained within the second RT coil (RTC)
Perhaps RTC1<RTO2> where the < > symbols are used to indicate 'contain'- i.e. Round Turn Coil No.1 contains the origin of Round Turn No.2
So the whole Myrtle could be expressed as RTC1<RTO2> // RTC2<RTO1>
Or perhaps more descriptively :-
Myrtle : RT(2) : RTC1<RTO2> // RTC2<RTO1>
Telling us that its name is Myrtle, it contains two round turns and that they are configured such that the coil of each contains the origin of the other.
On that basis can you make out what this description is:-
Bowline : RT(1), B(1) : RTC(1)<BL(1)> // BE(1)<RTL(1:1)> // RTL(1:1)<RTL(1:2)>BL(1)
{key BL = Bight Lead; BE = Bight Eye; RTL = Round Turn Lead and RTL(1:2) = the second lead of the first Round Turn}
Note:- RTL(1:1)<RTL(1:2)>BL(1) might need a little explanation - <RTL(1:2)> the second lead of the first Round Turn is contained between RTL(1:1)< the first lead of the first Round Turn and >BL(1) the leads of the first Bight. As there is only one round turn and one bight, it is perhaps easier to read the formulae by leaving out the component number i.e. RTL(:1)<RTL:2>BL - the Round Turn lead 1 traps Round Turn lead 2 against the Bight Leads.
Have you started something here Patrick, or has it already gotten to be too complex to have any value and even at this complexity will it fail to describe more complex knots? ? ?
Derek
