Nautile aka Charles Hamel's personal pages
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Always beware of what you see on a 'maker' documentation



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.

Behind each of these words hide different phenomenal facts.

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

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 ?

I seriously doubt it as real-life shadows do not discriminate the nature of the crossing. 

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 ?



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)

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.



By mixing the two preceding tools 
you get another one for free which works a bit differently  from its "parents":  "Shadow-Tangle" or "Tangle-Shadow".


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

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