Instead of 'one strand' it can be a
them in strictly parallel courses as in doubling
a case they
following the same LEADer
It is 4 LEADs in the example given - note that each LEAD
"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
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
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
inside and five outside.
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
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
on a map.
In fact it is *not* really a sinusoidal curve.
The number of BIGHTs is the number of 'inflexion' or
of trajectories, either at the
(outer) border or the lower (inner) one, before beginning to be obliged
leader", that is to double the THK.
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
MY IRREGULAR TURK'S HEAD
One day I was playing at enlarging TH and happened
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
In that it is 'irregular' : not obeying the rule and *not being a THK
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
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
TURK'S HEADS KNOTS (THKs) ARE NOT BRAIDS
a full dissertation in .pdf format that was published by Knot News
letter #65 and #66 ( 2008 February )
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
on not too shaky grounds.
Granted ! every knot can be made into a closed
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
at one of their extremity
aligned parallel to each other (
make a braid that is what knotters do)
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
braid already laid. A given
only allowed to change direction following a lateral vector
immediate neighbour(s) so as to form a
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
digits, it give meaning to other braids ), are fixed( or
May be it will feel a bit less 'convoluted' if you imagine a
vertical group of strands equally
spaced from each other two by
They are suspended by their upper extremities, pulled downward by
own weight and
are only allowed to
modify their course by going laterally so as to cross one of
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.
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
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
cylindrical (cylinder of
revolution), or circular in case of a mat,
THK the result will be
is indistinguishable from a
apparition of a discontinuity in the
STRAND that makes the THK transforms it in
Those second order
STRANDs make the
braid, the braid effect.
If you need to "destroy" the THK to make the "braid inside" it appears
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
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
morphology for a
not obeying the "warp-weft perpendicularity" point when and
only when the knot is
finished but in process it is an interlacing,
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
Bruce GRANT at
All this is of no real importance in the run of the day way
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
Morphological ordering on
external appearance based on a postulation : degree of 'likeness
appearance' is equal to degree of genetic kinship is a risky method
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
aspect" as criteria ?
IS ANY CIRCULAR KNOT OF ANY COMPLEXITY IS A TURK'S HEAD
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.
...to be developed later...
MY (personal and provisional) PROPOSAL TO STATE
WHAT A TURK'S HEAD IS
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
cutting it on the appropriate course
result in a different knotting
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
Copyright 2005 Sept - Charles
Hamel / Nautile -
Overall rewriting in August 2006 .
Copyright renewed. 2007-2012 -(each year of existence)