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BIGHT / LEAD counting as I understand it

No warranty given !

LEAD = (sometimes it is 'part' or 'strand' or 'turn' ) the number of different 'strand' you will
get if you do what I named  "a braiding transform".
A braiding transform is cutting the TH following the shortest route  between the borders :
the route  perpendicular to the two rims.

Instead of 'one strand' it can be a group of them in strictly parallel courses as in  doubling
or tripling. in such a case they are all following the same LEADer

It is 4 LEADs in the example given -  note that each LEAD is "doubled" but that counts
 as 'one cut' only as it is a strictly parallel trajectory that is followed by its two components.
Consider them glued or fused as one big enlarged strand if they are strictly parallel one to
the other(s)

Here you have a "tripled" or " double doubled"  or "twice double" cylinder TH with
About LEAD doubling here is a simple explanation .
Alternatively imagine you are using a different colour for each lead.

Still alternatively :  count "crossing" or "hole", following the shortest zig-zag route from one
knot edge to the other, taking out of the counting the external crossing BIGHTs make by
crossing each other.

How many 'snaking' courses you cross by cutting direct from one bight on one border to
its direct homologue on the other border :  that is you number of LEADs.
Similar to the problem of post and interval in a fence, here it is 6 "interval" so 6+1=7 'post'
or rather LEAD.

BIGHT (can be scallop, cross) : count on one border or one rim only all the arciform scallops.

5 BIGHts it is as there are 5 colours inside and five outside.

It is easier still to do the counting with a cylindrical TH .

This will work unless you have a  Moebius / Möbius strip (that some being egregiously
mistaken and ignorant confuse with a  turk's head knot - in which case you deal with only
one border instead of 2 ; only one 'face' too. )

Do you agree ?  Is it clear for you ?

To visualize try to see the LEAD course as the sinusoidal projected course of a satellite
on a
map.  In fact it is *not* really a sinusoidal curve.

The number of BIGHTs is the number of  'inflexion' or 'curving' of trajectories, either at the
upper (outer) border or the lower (inner) one, before beginning to be obliged to "follow
the leader",  that is to double the THK.

Summary :

The easiest is BIGHTs counting :
Use only one border and count the number of "inflexion" or "bump".
Those that are not "on the frontier" are not BIGHT in their own right and belong to so called
 'inner bight' category.

LEADs counting is a bit trickier but still easy with the "braiding transform" trick.



One day I was playing at enlarging  TH and happened to forgot the first correct move of
ENLARGEMENTt and got this 'impossible' TH

I write 'impossible' because it does not follow the rule of the relation between number of
bight and number of lead= it has 5 bights on any of its 'borders' and 5 leads.
In that it is 'irregular' : not obeying the rule and *not being a THK really.

But there are 2 'internal', 'inside' bights (so that would go as 7B and 5L which is would
obeying the relation.)
If a cut were to be made from one border to the other  the destruction of the turk's head
knot will make it into a true braid, and this one would resolve into 5 leads braiding effect,
each of these 5 strands being doubled.


a full dissertation in .pdf format that was published by Knot News , Igkt-PAB
letter #65 and #66 ( 2008 February )



and for the fun of it : the tale of the Chess Knight and the THK

This is with a view toward a 'classification' of knots

I know that some, among whom are better brains and better knotting hands than mine,
think and write about them as "braid" but I beg to differ, and that I believe I am allowed to
do on not too shaky grounds.

Granted ! every knot can be made into a closed circular/cylinder braid, but this is
 mathematics point of view and I want to leave topology aside here and write only about
the real knot.

It is clear to me that SINGLE STRAND (even multi-strands one) THK are not braids in
their process of construction even if, once finished if you accept or except the circularity or
once destructed,they can be mistaken for a braid.

Still that does not make them "braid / plait ".

This is braid effect as Ashley recognized a sinnet effect in
[open quote]
#1685. The sinnet effect is also in evidence in this one……
#1686. In this knot the sinnet effect is carried still farther…
[end quote]

Though not as strict as mathematics do here goes a definition of a braid :

a braid is a set of strands (N>=2) held equidistantly fixed in place at one of their extremity
aligned parallel to each other
( operationally to make a braid that is what knotters do)  before the
first crossing is made.  They follow a constant general course, along which no strand is
allowed to back track making it that no strand can cross itself or follow again a previous
segment of the braid already laid.

A given strand is only allowed to change direction following a lateral vector toward its
immediate neighbour(s) so as to form a crossing between each other two by two, either
High or Low ( H & L  or  Over/Under).

( strandS : plural as the null braid of one strand is of no interest here even if as "zero" gives meaning to
the others digits, it give meaning to other braids ),
are fixed( or in the case

May be it will feel a bit less 'convoluted' if you imagine  a vertical group of strands equally
spaced from each other two by two.
They are suspended by their upper extremities, pulled downward by their own weight and
are only allowed to modify their course by going laterally so as to cross one of their
immediate neighbours.
Never would they go upward or follow a segment of the braid already set.
So it is an impossibility that any one of the strands can cross itself.  
This interdiction of self-crossing is a major point. (a spliced eye is not a 'braid').
When the braiding or plaiting process is finished the final move is to fix in place the lower
extremities of the strands.

If this general description of a braid is accepted it follows that it must be accepted that a
THK, even multi-strand, cannot be considered to be a braid or a plait.

It is a point to note that at the cost the destruction of the cylindrical (cylinder of
revolution), or circular in case of a mat, THK the result will be 'something' that
is indistinguishable from a true braid.  

The apparition of a discontinuity in the STRAND that makes the THK transforms it in
several different segments.
Those second order STRANDs  make the apparent braid, the braid effect.

If you need to "destroy" the THK to make the "braid inside" it appears then to my mind this
shows that the THK is not a braid.

Additional notes :

---part one :
THK are not 'weave' either if you  consider that in the making of a weaving the warp is put
in place before any weft is put in.
Warp and weft : that makes for two strands or set of strands.
Even if THK could be a weave in the case of the 2S  that would not be the case for knot
going under the name of THK with a number of strands used in their making differing from 2.
Warp and weft are perpendicular to each other, that is characteristic. 

I will admit that 2 STRANDs THK may be (mis)taken in their resulting external
for  a weave not obeying the "warp-weft perpendicularity"  point when and
only when the knot is finished but in process it is an interlacing, intertwining, ordered
intermingling, not a weave.

---part two:
A part  of what 'braiders' present under the label 'braiding / plaiting' is not braiding but
another type of interlacing or ordered intermingling of strands.
See Encyclopaedia Of Rawhide And Leather Braiding by Bruce GRANT at Cornell
Maritime Press.

All this is of  no real importance in the run of the day way of putting things but if  one day
 'knots" are to be correctly ordered in "relatively homogeneous groups" then it is really
necessary not to put things that come to existence by quite different processes under the
same label for the motive that " they looks alike".

Application of morphological criterion is very bad 'phénétique'/ 'phenetics'.
Morphological ordering on external appearance based on a postulation : degree of 'likeness
in appearance' is equal to degree of genetic kinship is a risky method which would  for
animals lead to put the thylacine (Thylacinus cynocephalus ) with the mammalian wolves or
the whales and dolphins with fishes.
What will you do with the 'ornythorinque'/'platypus ( Ornithorynchus anatinus )  if  you
chose "external aspect" as criteria ?



In my view abrupt answer is no.  

Or you have to admit that the circular (so perfectly so that it is a sphere -need a core ;
without core and shrunken on itself it is a ball ) Monkey's fist is a turk's head knot as I have
seen some do. be developed later...


(personal and provisional) PROPOSAL TO STATE

I fear that my training took the best of me : the best assured diagnoses are the
post-mortem, forensics ones.

Provisionally , at this moment in time :

A turk's head knot is a knot that, when destroyed by cutting it on the appropriate course
result in a different knotting that is morphologically undistinguishable from a braid/plait but
was made using a process different from braiding.
My personal interpretation of the apparition of  "braid" and "laid strands" in the mind of Man
is that it stem both from the observation of Nature and from playing with long hairs
(remember that grooming play a very important part in the life of social apes of which we
are, even if we tend to forget that and put ourselves apart from Nature with much hubris.


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

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