Nautile aka Charles Hamel's personal pages
Page1   Page 2  Page 3   Page 4  Page 5  Page 6    Page 7   Page 8   Page 9   Page 10
Page 11   Page 12   Page 13  Page 14   Page 15  Page 16   Page 17     
Page 18     Page 19  Page 20  Page 21   Page 22    Page 23   Page 24   Page 25   Page 26


I have added quite a number of pages to my work on THK mathematics to use a big word .

I would very much like to verify their validity against SCHAAKE 's works that I could not got hold of :
books and pamphlets ( not "The Braider" magazine that I can get ) at the lowest cost possible , DVD or
CD , paper printing are not necessary. This in 2008September.

or keeping logical planes well separated :

structure,or finished fixed state
process, ephemeral phase disappearing into the fixed state.

I was set onto this trail years ago by Tom HALL's book :
 "Introduction to Turk's-head Knots" in which he succinctly evoked the CYCLE (I do prefer "PERIOD" if you don't mind).

Unfortunately the work from which he got his idea that he so sketchily exposed without any
"justification made in a reasoned fashion" has stayed unavailable to me at the moment of
writing.(2008 Sept).

In a lifetime worth of experiences I find that a most common intellectual mistake is
confusion of logical planes.

Discussions about knots are rife with that if I must add credence to my experience.

Present topic is an echo of THK ARE NOT BRAID .
Only a dull teacher will think that they are"best described as continuous braid"for beginners.
Beginning a teaching by teaching a mistake is certainly not good teaching.
"continuous braid" is an utter nonsense, as if a braid is not a continuity ! plus they are NOT braid ANYWAY as clearly demonstrated in my paper.

I hope I made a convincing demonstration in it that there exist essential differences
between a BRAID and a TRUE THK ( THK or NOT THK) which do not, repeat, do not
share the same essence as their geometry show perfectly when studied.

In the space-time distribution of the route(s) followed by the strand(s) in the process of making, each sort (braid / THK) is indeed quite different from the other if you are not blind to the visual evidence, or deaf to reason. Dumb I will not pronounce about!
No one is that dumb anyway in the knot tyers community. ( in case you missed it blind - deaf -dumb (in a different meaning ) stand for the 3 monkeys of wisdom ;-)  )

For structure recognition the STRAND's route is more important in a knot  that their crossings are.  

In fact the strand is following its route along the SHADOW of the knot and in the actual
knot it obey the crossing sequence program (as in my H & L system) or the so-called
That is my conjecture  in the mathematical sense of the word : something I recognize -and
verified as true countless times- but cannot formally prove)

The sequence of crossing built by reading their order of appearance when following the
SPart-WEnd vector in a *finished* knot cannot really be of any great use to immediately
recognise a structure at a glance.

Please take good note that the two sort of sequencing are hugely different  : order in the actual making following the diagram given by the SHADOW or CORDAGE ROUTE and
order read on a finished knot and re-tracing the cordage route )

Just to show you :

What is the one you tentatively identify ( can put in a broad class )  in a thousandth of a second   
or  this "cordage route"  without any nature of crossing represented which make it in fact
equivalent to the concept of SHADOW, taken from topology,  exposed elsewhere in these pages.

Second try ?
or this route without any crossing type mentioned. (think SHADOW)

I will bet that every one was much quicker recognising the class of the route or SHADOW
than in identifying with a knot's name the sequence of Os/Us.

Recording  as my H&L system is supposed to be used for) the order of their appearance
on the route followed during the process of laying the knot is in fact as much or may be
more a recording of the route ran over by the STRAND than of the crossings per se.  

The route alone, without any type of crossing being specified, but just there mere existence
being noted (SHADOW) , gives more information about the "class" of the knot ( I will say
out of hand 80% at least) than the crossings themselves taken in isolation (I will say 20% at

There is less degrees of freedom in the crossing sequence than in the route ;  my guess is
that it is one of the possible explanation of the above.

A knot is more swiftly assigned to a downsized  probable group of appurtenance by looking
at its SHADOW (the diagram of the cordage route)  than by being given the visual sequence
of the crossing in the finished knot.

BUT this has a big downfall : lazy minds will stay there and believe that identical shadow
cover similar knots. How false that can be

No wonder that so many are still erring in obscurity. ;-)

I suppose that the shadow is the explanation why obdurate brains still persist with the
greatest energy they can deploy in negating facts and data and stay with the illusion that the
"THK umbrella"  (to steal what I find to be a very funny expression) cover all sort of
NON-THK knot.
This is a case where creed born from blind adhesion (unexamined by sender and by
receiver) to handed-down recipes (gospel if you like) reinforce itself at each
presentation of facts (the heathen miscreant's words for them The True Believers) showing its mistaken aspect.

The bright ones will only have been in  a transitory phase (which must never be extended to
become a permanent state), called "error" and have mistakenly confounded the shadow with
the complete knot nature for a moment. Thanks to having an open-mind in autonomous (as in not-programmed and not-blocked by creed)  thinking mode they can change tack and built new knowledge.

[ illusion is an inability to get the correct perception that will persist even when warned about
it : it is just a defect of the system that is quite unable to get the true picture!

 Whereas an error is simply insufficient data having been furnished and or analysed so that it
could be built into sufficient information to become acquired knowledge ; simply give or
point to the data and the correction is readily made by personal thinking]

The 'gestalt' of the knot lies in its SHADOW or "CORDAGE ROUTE" and
certainly not in the nature of its crossings and their sequence

One "proof" I can propose that the cordage route (SHADOW) is much more important
than  the crossing sequence as recorded, not as by my H &  L system along the way of the
making, but on the finished knot, following the SPart-WEnd vector and tallying the crossing
as and when they are encountered is the following fact follows :
while NO CROSSING has been altered in its nature, no rethreading was done, we can
have 2 really physically different knots that the topology recognise as being equivalent ( which is quite a bit different from "identical" ).
Unchanged crossings but different cordage route.

This the case of the Fig-9  and #925 that I solved and show in another page.
picture one - picture two -picture three - picture four

No crossing physically re threaded and yet two quite individualized knots :
The SHADOW (cordage route) is more important than the crossings as usually
noted in the order they are met in the finished
knot. (quite different from the
sequence of crossing as use in my H & L system which is in fact the "crossings
route" so to speak )

So here we will attach ourselves to distinguish  between :

--- the order of BIGHTs as they can be seen in the finished knot (in the illustrations  these
number are denoted by Roman digits, these Roman digits though looking to be an "ordering"
variable are in fact - we will see why latter -  mostly mere "nominal" variable.)


--- the order of BIGHTs making during the process of laying the knot (denoted by
Arabic digits). This is not a "nominal" variable but a real "ordering" variable.
Point "zero" is not arbitrary ( even if reasoned) as in the first sort of ordering evoked immediately above.

THK 7L 5B has  

 [ I ; 5]  [ II ; 3 ]  [ III ; 1]  [ IV ; 4]  [ V ; 2 ]    OUTer ring
 [ I ; 2]  [ II ; 5 ]  [ III ; 3]  [ IV ; 1]  [ V ; 2 ]    INner ring
 the circular permutation , one among several possible, is as they say in French " visible
comme le nez au milieu de la figure" , "visible as the nose in the middle of the face".
5 3 1 4 2
   3 1 4 2 5
      1 4 2 5 3
         4 2 5 3 1
            2 5 3 1 2  
and we can still do it in the "reverse" !;-)  2 4 1 3 5
                                                              5 2 4 1 3
                                                              3 5 2 4 1 ... I trust you have got it by now.

the 3L 4B has  [ I ; 4]  [ II ; 1]  [ III ; 2]  [ IV ; 3]  OUTer ring
                       [ I ; 3]  [ II ; 4]  [ III ; 1]  [ IV ; 2]  INner ring

There is a phase offset between INner and OUTer.

NB : of course depending on the way (Clockwise  or Anti or counter-clockwise)  you
make a drawing of a closed curve or on the way you distribute the BIGHT OFFSET ( L/2)
in cylindrical diagrams between upper and lower) the under laying sequence stay identical *but* the permutation you  get immediately visible is different.
So you better standardise your practice so as to keep things comparable and

See the effect of the 2 different distributions of STEPs among TOP and BOTTOM RIM.

5L 4B  so STEP L/2 = 5/2 = 2.5  so 2 and 3 must be use as you cannot be floating
between 2 pins.

Two ways :
2 on upper  3 on lower 
3 on upper 2 on lower

This is not really important as long as you take great care when in transaction with other
persons to make ( *and* keep)  your choice EXplicit and not IMplicit .
Be consistent and congruent all along your discourse : always use the same frame of
reference knot diagram after knot diagram.(or warn the other person you are changing it )

Personally I use :
-- for the spirograph drawn curve : CW if RC (Rolling Circle)  is signed minus and CCW if
signed plus.
-- for the cylindrical diagram : I offset to the LEFT the numbering of the BOTTOM RIM
and apply the smallest ( if this apply) of the L/2 to the top rim. (this may change with time)

Say that 5 so L/2 = 2.5 which mean using 2 and 3, then I use 2 on upper and 3 on lower.
So the lines moves from  BOTTOM-RIGHT to TOP-LEFT and
                             from TOP-RIGHT to BOTTOM-LEFT .

For the 7L 5B  in fact the structure outer rim order  I  II  III  IV  V is in correspondence
with the process order     5 3 1 4 2  

5 3 1 4 2  is  4 2 0 3 1  if you start with 0 instead of 1 - remember your early days at
elementary school : to count  items you can either count  them 1 to n or 0 to (n-1)

That just show the full importance of always minding about NOT mixing structure and
process : the route is not to be equated to the crossing sequence or the manner of
tying per se !

The manner of tying a knot is an epiphenomenon only, that has absolutely no value to
classify or distinguish 2 knots that could be qualified "different" just because of that
difference in the manner of tying them. In other word : there can be several manners of tying a given knot : several recipes arriving at identical structures ).
Their common route in the finished knot is much more important that the mean of
obtaining the final common route

That is specially true if you want to compute a "distance" between two different routes for a

Classification of knot is probably one silly endeavour, silly because it can be equated to
emptying the seas and oceans using a pail without bottom !

The best (IMO) one can hope is, one of these days, to find a way to conceived a fast
identifier of knots to discriminate between them.

So how can we, without actually making the knot, know in which order the BIGHTs will
appear during the making of the knot ?

There we need the help of a bit of mathematics that will be seen latter.

Before tackling that it is better to absorb, digest and assimilate the full of  MATHEMATICS
and THK
Part One     Part Two
Those 2 Parts and what has been said elsewhere in those pages.
This is the absolute minimal needed I think.

A tip : if need be read again mainly Page 3, but also  Page 2, in particular the "corridors" use
and the PINs JUMPs or the PINs STEPs then think by yourself  about how to get the
processing order of BIGHTs when knowing the L & B of your THK.


NOT TO BE DISPENSED WITH if you don't already have a working  grasp of
what MODULO / MODULUS  is :

Persons wanting a good intro  : just go over there
Some quick formulas

I urge you to go to :  ( FREE )
and use get a feeling

Here by modulus I mean the measuring stick

Without understanding the use of modulus, THK anatomy and making cannot
really be
understood in depth, even by the very best practical first order practitioner of the genre. (some among the best are really lousy at theory !)

That does not mean someone applying the handed down recipes will not produce stunning
I can drive my car without being a mechanics or an engineer but I would not boast of
"understanding" it in a profound sense.
A spider can do stunning work without a brain, with just a good innate program,
cockroaches can display highly complex behaviour (and adapted to survival too) using
only a few nervous ganglia, octopus do feats in  recognition of patterns and are quite adept
at "manipulation" err  "tentaclation" ;-)

I just happen to believe that to be a knot tyers is to be using one's available brain as much
as possible in order to truly study and understand the deep nature of what one is making
and not simply to have agile hands and fingers and myriad of "unthought about "programs
(recipes) for knots.

Consider those 2 series of integers:
ODD parity  -8  -6  -4  -2  0  2  4  6  8
EVEN parity  -7  -5  -3  -1  1  3  5  7

modulus 2 is used to built each one,

but this mean too that you can represent the 'ODD' series with '0' and the 'EVEN' series
with '1', in other words with their modulus 2 ( any 'EVEN' number divided by 2 has 0 has
remainder and any 'ODD' number divided by two will leave 1 )

A PERIOD (cycle) is a modulus operation.
A bit like a counter starting anew from 0 once the extremity of the modulus has been
Like a clock measuring the whole day-night span which by convention finish at 24 which is
also the beginning ( 0 point ) of the next period.

With modulus 12 you will use 2 (mod 12) to mean that it is 2 AM or 2 PM.
This leaves some ambiguity but for a 24h nycthemeral cycle there is the need to specify the AM or PM part of it.
Using modulus 24 , 14 (mod 24) will without doubt state that it is in the afternoon so
equivalent to 2 PM.

If you know how to perfectly read and use a mechanical watch then you are making use of
modulus, even if you never realised that.

When, in the end of the afternoon,  at 6 PM you say that it is 18:00 then you are making use
of modulus 24. while with 6 PM you are using modulo 12.

When at 22:00  Saturday evening you decided to make your clock ring in 8 hours, you can
immediately know that it will be ringing the next day, Sunday, early morning at 06:00.

22:00 + 08:00 = 30:00
but 30:00 is way over the 24/0 hour mark
so 30:00 minus 24:00 = 06:00 or 30 divide by 24 than is 1 for the Integral Part and 6 for
the Fractional Part. Here you keep the fractional part.

( beware this is "correct" as far as results obtained go but is light years from mathematics
formalism ; but then seeing what wonderful work the great Einstein made explaining relativity
in Principles of Relativity first English edition 1920 (1952)  book (read it it is a marvel of
intelligence that makes you for a while feels you are intelligent).
I feel that the tracks of absence of formalism for the sake of clear understanding was well traced and alas I never will begin to 'reach his instep, much less his ankle' as they say in French )

Imagine you own a measuring stick which markings only appear after it has measured the
whole length that is to be measured,  leaving no remaining length to be measured (zero) or a
length smaller than the stick.
Modular is like those very good cook who are past master in the art of making something
with the "left over" ;-)

If you are measuring something "negative", say the part of a partially build wall till to be built
as measured starting from the point the wall will attained when finished, you have to measure
a part of the already build wall at the end of your measurement of the empty space.
You may not stay in the "minus", you cannot rest your measure below the zero mark, you
have to climb over it..
Your measuring stick must, somewhere, somewhat,  touch the built part of the wall.

Then you have used of the modulus concept, the stick being the modulo unit.

5 modulus 3  has 2 as result     { 5 - 3 = 2  stop here as 2 < 3 }
8 modulus 3  has 2 as result   { 8 - 3 = 5    5 -3 = 2  stop here as 2>3
-10 modulus 3 has 2 as result  {-10 + 3 = -7  -7 + 3 = -4    -4 + 3 = -1  we cannot stop
here even if |-1| is smaller than |3| we must go over zero    -1 + 3 = 2   we can stop here
so we can state
5 == 8  == -10  modulus 3, that is  5 == 8  == -11  "in the frame of reference modulus 3"

You measure what is to be measured with length of stick after length of stick till there is
either nothing left (zero) or the length left is less than the length of your measuring stick.
The number of  full length of measuring stick is discarded.

If measuring a "negative" then you go on laying length of stick after length of stick till you are
either touching the zero mark or  have gone beyond.

If modulo is 3 then:
(-3)  lead to result 0       (+3) lead to result 0
(-4)  lead to result 2       (+4)  lead to result 1
(-8)  lead to result 1       (+8) lead to result 2

Copyright 2005 Sept - Charles Hamel / Nautile -
Overall rewriting in August 2006 . Copyright renewed. 2007-2012 -(each year of existence)

Url :