Always beware of what you see on a 'maker' documentation.
SHADOWS CHASING !
I will try to show why I do not like
indiscriminate use of : 'SHADOW', 'SILHOUETTE',
'OUTLINE', as if one could be taken as equivalent to any of the other,
as
synonyms.
I prefer 'SHADOW' as it entails 'a projection' and I will make use
of that notion later.
More precisely projection on an oriented horizontal plane.
So I will use 'SHADOW', being understood that this shadow while not
showing the true
nature of crossings ( High or Low ; Over or Under),
still show every
one of these crossings
once and once
only.
Never will one crossing be superposed on another one.
At any one place there can be only one crossing
projected, be it in a 'shadow' or in a knot
diagram.
By the way it is an affirmation of mathematics : any knot
can be drawn in a diagram.
Problem is just to find the right way to do it !
Shadows
can be used to 'group' or 'separate' knots.
Under one shadow several different knots can hide.
This is how
to proceed to draw them.
This is my attempt at a
depiction of the concept
using real life shadows.
Would you have been able, just by looking at the shadows to
discriminate
between the
Fig-8
and the
Non-knot ?
This is what make them useful to study
knots while trying to find 'group' to be use
in
ordering, cataloguing...
Remember I still hope that cladistics
principles and methods can be 'adapted' to knots and
'adopted' by the
knotting community.
Thinking this is a bit tangled ?
TANGLES
Be not in fear, I am not a mathematician, I
will not make use of the full
mathematical
concept invented by John H. Conway.
Tangles can be computed, what we will not do.
Still this is a good
tool to study 'real' knots.
There was a Rubik's cube with tangles instead of colours.
This free software draw them.
A
tangle is a knot projection (or a link projection, entire
knot or part of it)
on a plane,
surrounded by a circle (I also use
2 bars as in the previous paragraph) and the knot or link
crosses the circle exactly four times, say in the four orientation NE,
SE,
SW, NW.
I will depart from the formalism of this idea and use 'tangle' with
more than 4 parts.
Think sheepshank as an example.
One can view the four crossings as 'fixed', and with no particular
identification as tail /
working end (WEnd) or Standing part (SPart) /
Load.
This absence of identification can be really useful in studying
the knots anatomically as
tail / working end (WEnd) or Standing part (SPart) /
Load are 'physiological' (so to speak)
notions.
I find"tangle" a useful tool .
This
introduce a "cross-breeding" notion : "shadow-tangle"
The notion of 'equivalent'
tangles
is a useful
tool to group or separate knots.
I am still hoping to compute 'distance' in
and between 'groups'.
It would be nice to design 'unit' tangles, a sort of alphabet of
tangles
to draw knots.