GEOMETRIC TRANSFORMATIONS
of knots diagrams and of real 'in volume' knots

Just think about the accumulation of so much particular points that it seems that UK English, Canada English, USA English, Australian English seems at times four different languages!

Not easy to be "international" for you born to one of the English-English based language but think about us "foreigners"!
At times I find Middle-English easier that following the particularities of USA or Canada or Australia.

Knots (mathematician's, diagrams or real ones) are topological and geometric entities and as such the words used to speak about operations on them that modify their geometry
should be very strictly defined and the use of words every day meaning SHOULD BE
AVOIDED
as much as possible in a technical discussion.

A highly 'guarded' or surveyed language is a compulsory condition if any ambiguity or even plain non-sense are to be avoided.

Euclidean transformations preserve LENGTH (linear dimensions) and ANGLES so the shape will not change ; only its POSITION and ORIENTATION will.

Note that either the geometric object is transformed or it is the coordinate system which is. This is about transformation of the geometric object itself.

We will deal with the 2D, plane projection of knots and with the real life 3D knots.

First some depiction of some image manipulation.
Image manipulation with a computer does not give any modification of crossing as one get with a real cordage or as can be done when drawing it.
So do not use the " flip" effect believing - it is a quite common egregiously mistaken belief - it will be a real "flip" as is turning a page in a book : you will get the same result as seeing the drawing in 'transparency' through the paper and not the actual view of the 'other side" as when 'flipping' a real knot.
This is quite important to keep in mind.

In fact it is not MANIpulation, with the hands, as all this is "thought experiment", with
Real Knot In Real Cordage you would have to thread anew the whole knot to get the
material result that 'image manipulation' gives you.
Nevertheless this is not to be discarded lightly.

Axis setting.

I do hope I did not trip in the mathematical carpet here.
Please make any dreadful mathematical error known to me ; my University years are way behind me.

Small note to avoid further muddling : I strongly think that it is inadvisable to mix words coming from another domain than geometry. The more so because most often  they are both unclear to the lay person and because it is best to stay coherent : e.g using 'converse' as in 'conversion' (which need the use of contra position or obverse) is not congruent with geometrical entities but only with logical entities. E.g of 'converse' : If P exist then Q does not exist / If Q exist then P does not exist.
Conversion must obey strict rules to be a 'valid move' and is a quite unnecessary complication here. Une variation élégante ! only

2D or PLANE

 Transform ident In 2D plane before After remark SCALING ( x , y ) ( x.sf , y.sf ) sf : scaling factor  this is how you get SIMILAR TRANSLATION ( x , y ) ( x+tf(x) , y+tf(y) ) tf : :translation factor, just a displacement without resizing. Accept SIMILAR BACK view ( x , y ) (- x , y ) it is left-to right or (x , -y  ) in second case it is upside-down change tail and SPart are as in the 'original' vector but it is the "other side" that is now the 'obverse'  side. A 'reverse' with its own name. REVERSE ( x , y ) ( -x , y ) or (x , -y) tail and SPart are interchanged from the disposition in the original It is coming back following the course previously followed. INVERSE ( x , y ) ( -x ,- y ) On a plane it cannot be differentiated from REVERSE MIRROR ( x , y ) ( -x , y ) or ( x , -y) depending on the axis used. Inversion of DEPTH ( -x or - y) only IDENTICAL ( x , y ) (x' , y' )   ( x+n , y+m ) either in another set of coordinates with unchanged units or without changing the set of coordinate and using translation.for instance ( no resizing occur) SAME ( x , y ) ( x , y ) set of coordinates are unique in both case so same apply only to 'original' ROTATION ( x , y ) ((x.cos alpha-y.sin alpha), (y.cos alpha-x.sin alpha)) alpha : angle of rotation  -  Accept SIMILAR

3D Cartesian coordinates
 Transformation ident In 3D before After Remark SCALING ( x , y , z ) ( x.sf , y.sf , z.sf ) sf : scaling factor  this is how you get SIMILAR TRANSLATION ( x , y , z ) ( x+tf(x) , y+(tf(y) , z+(tf(z) ) tf : :translation factor, just a displacement without resizing. Accept SIMILAR BACK view ( x , y , z ) ( -x , y , -z ) or ( -x , -y , z ) or (  x , -y , -z ) tail and SPart are as in the 'original' Changing only 2 axis so as to view "the other side". A 'reverse' with its own name. REVERSE ( x , y , z ) ( -x , y , -z ) or ( -x , y , -z) or ( x  , -y , -z) tail and SPart are interchanged from the disposition in the original. It is turning back the way that was previously followed. Changing 2 axis INVERSE ( x , y , z ) ( -x ,- y ,- z ) Changing all three axis at once MIRROR ( x , y , z ) ( -x , y , z ) or ( x , -y , z ) or ( x , y ,- z ) depending on the plane used. It is inversion of DEPTH ( -x or -y or -z ) only IDENTICAL ( x , y , z ) ( x + constant , y + constant , z + constant ) either in another set of coordinates with unchanged units or without changing the set of coordinate and using translation for instance ( no resizing occur) SAME ( x , y , z ) ( x , y , z ) set of coordinates are unique in both case so same apply only to 'original' ROTATION ( x , y , z ) (  x.cos alpha -y sin alpha , x.sin alpha+ y.cos alpha, z ) alpha : angle of rotation - Here I will stay on rotation about the Z axis "around" it, the knot diagram being on the x, y plane and Z axis being perpendicular to it. x.cos alpha -y sin alpha , x.sin alpha+ y.cos alpha, z

REVERSE looks to me very ambiguous in its every day use.

It can means :
- 'back view'
or
- it can means turned backward like 'ab' being 'ba' or as in 'reverse gear',
- it can mean 'in an opposite manner'.
It can be use as 'opposite or contrary', or even "the back of.."

So please do no use 'REVERSE' about knots if not applying the meaning of the geometric transformation.   [  (x, y, (z))  becoming  (-x, y, (-z)) ]
That will makes for a clearer discourse.

Better not use REVERSE about knot except with this meaning :
in an opposite orientation SPart/WEnd and keeping the crossing where and of what nature they were in the 'original'.
In particular it is highly faulty IMO to use REVERSE to mean MIRROR as Ashley and many of his followers were, and still are not shy to do.
Flip a real 'in the rope' knot and you get a real back view of it, Flip a 'representation' of it,however precise it can be, and you will only get the equivalent of 'mirror' : imagine drawing your knot on a piece of transparent glass and flipping the glass, that is equivalent in result to the flip on the screen of you computer.

In real 'in the rope', to reverse , actually reverse a knot, is  to revert it to an unknotted piece of cordage.

The 'reverse' of an oriented knot ( from SPart toward WEnd ) is simply a  knot  positioned elsewhere ( not same knot please !) with unchanged crossings but with the 'opposite' orientation of SPart / WEnd.

It is mentally leaving the knot unchanged in its crossings and exchanging the WEnd for the SPart and conversely or vice versa.

When in a knot the WEnd is made to follow on a parallel course the 'first laid'  it is obviously not making a 'reverse' but making a 'rethread' , so making another sort of knot.
This is true too when you put the WEnd in a "slipped" fashion.

Reversion is also meaning right-to-left reversion of an image.
So as a plane mirror never put left where right is and vice versa reverse cannot be use in place of mirror without using a faulty wording.

INVERT : in every day language the word  may be understood as 'turning it inside out',  like what can be done with a glove or as 'putting it upside down' as with a glass.

Speaking about knot INVERSION should be strictly restricted to what was exposed above.(x, y , (z)) becoming (-x, -y, (-z))

I put there an illustration in coordinates of an inversion.

As for the 'inverse' of a knot it is :
- first a 'reverse'
- then reflection/mirror image of this reverse.

So INVERSE = REVERSE + MIRROR

INVERSE is REVERSE ( exchanging the WEnd function for an SPart one and
vice versa ) PLUS changing the nature of each crossing ( a High / Over becomes
a Low/Under and vice versa)

So please be careful with 'same', 'reverse', 'inverse', 'mirror', 'similar', 'identical', 'comparable'....

When 'inverse' is applied in conjunction with a rotation, it becomes an 'improper rotation' in geometry parlance.

I prefer not to dwell on 'invagination' or 'eversion' that are in use in my trade.

MIRROR IMAGE or REFLECTION : please use it to mean that for the image the sign of one and only one of its coordinate is reversed. ( see topic Mirror does not invert..)

MIRROR is not equivalent to REVERSE.

Two ways to get the mirror image
either flipping the tracing and keeping the crossings as they are in the original
or keeping the tracing as it is in the original but changing the sign of the crossing, reversing
the crossing BUT THAT DOES NOT MAKE A MIRROR A REVERSE !

ROTATION + INVERSION = ROTATION + REVERSE + MIRROR

SHIFT
A translation ( a sliding move ) without rotation or distortion

TRANSLATION
A transformation consisting of a constant offset with no rotation or distortion.
In a N-dimensional Euclidean space, a translation may be specified simply as a vector
giving the offset in each of the coordinates axis

TURN

Turn = a change in the course previously followed.

HALF-TURN
A rotation through 180 degrees (pi radians).

A turn along the x axis ( remember how I dispose the 3 axes) is an horizontal flip for
a software or back to front plus upside-down in every day life. This can be conceived
as a 'reverse' too.

A turn along the y axis is a vertical flip for a software or a back to front in every day life.

A turn along the z axis  is a upside-down in every day life. Problem is that it is too used to mean 'inverted' and 'inverse' is take to mean 'opposite' just as reverse' means 'opposite or contrary' in every day language.

Necessity should be clearly seen by now of an agreed upon ( not thinking that it is what I write that must be agreed upon !)  guarded , well defined, common language when speaking about those geometric entities that knots are.

DILATATION = EXPANSION + TRANSLATION

This is a SIMILARITY transformation : each line is transformed in a parallel line whose length is obtained by multiplying it by a fixed positive factor multiple of the original

A dilatation with expansion equal to one is simply a translation.

Of course you can do a dilatation with a null translation , it will be a dilatation about a central point, hence its name of 'central dilatation'.

A point that make disputes : similarity vs sameness

Falling into the temptation to say that either similar or identical objects are " the same" is falling into a logical fallacy : if there are 2 objects then logically they are differently spatially localized things and you may not use 'same' which refer to a unique individual.

If a particular knot, is submitted to translation, rotation,, in an Euclidean plane or space it will be admitted, for the sake of every day practicality, that it is "the same" though logically it is not.

BUT if you leave the 'original' in position and create a 'copy' of it ,even if absolutely perfect in any point then it seems better to say "identical" or if scaling intervened to say "similar".
Just as a 'clone' is not the 'same' individual as the original which has been duplicated, however perfectly.

Usage of 'Same' should be strictly reserved to cases when it apply to 'one unique'
individual and NEVER if there is more than one individual object/knot which are
referred to by the discourse.

'Same' almost never apply.
Frequencies are in favour of  'identical' or 'similar' as  the adapted word.
If you are loath to use them take a short cut and though this is logically faulty ( they
are indeed separable in the adjective applying to them ) use "indistinguishable" one
from the other.

Guarded word comes first and its every day equivalent is in ( )

ROTATION ( TURN )

REFLECTION ( FLIP )

TRANSLATION ( SLIDE )

When one apply any of those operation or a combination of thereof to a shape this
shape may become another one, in which case the 2 shapes are by 'construction' :
"CONGRUENT" but if you apply RESIZE then these 2 shapes are SIMILAR

NOW FOR CHIRAL - ACHIRAL - AMPHICH(E)IRAL

AMPHICH(E)IRAL : this is when a knot ( an object in the general case ) is fully
'reflexible'
- note that it is not 'reversible' - It means that it can be superposed to its virtual
image in a plane mirror.

Rather not use 'Handedness' which should strictly be restricted to living creatures
having hands.
Handedness a psycho-motor attribute about preference for which side of the body
to put to use.

Instead of handedness, in the other case ( e.g : sea shell, or migration off the eye in
flat 'bottom fishes") just use "Z" /"S" or "Indirect ( that is clockwise)" /  "Direct or
Trigonometric" rotation or migration....

Amphicheireal is  used to convey that 2 things would be identical EXCEPT (which
mean  that in fact, logically,  they are not identical at all !) for a mirror reflection.

For cordage use "Z" and "S" and nothing else ( be shy of left/right, avoid the dangerously ambiguous 'clockwise/counter-clockwise' and never use the all too often misunderstood "  with/against the sun).
About knots, " left" or "right", can be tolerated  if you are absolutely sure that you are
not introducing an occasion for misunderstanding through ambiguity of discourse.
Or use the signing of crossings or "Z" / "S".

CHIRAL : that is absence of mirror-symmetry between 2 forms .

Left trefoil and Right trefoil are mirror image one of the other.

( a parte : two 'molecules'' that are mirror image one of the other are what is called :
'enantiomers' or 'enantiomorph ; Note a wee bit of detail : our biology can only safely
use L-form and not D-form)'.
This notion could be adapted to the knotting world IMO )

Enantiomères/enantiomers are stéréo-isomères/stereo-isomers as it is the spatial
arrangement of the identical unit constituents ( global chemical formula is the same
in both case) that make the molecule one or the other.
You may use levogyre and dextrogyre for enantiomers.
Sea shells too are lévogyre or dextrogyre in the sense that they spiral one way or another.

Human cells can only function with proteins that are 'lévogyres'/levorotary, that is are
turning polarized light toward the left.

A dextrorotary molecule can be highly damaging to us while its levorotary mirror image is not ( THALIDOMIDE for sad example).

I feel that it is inadvisable to say that 'leftrotary, is counter-clockwise turn', a topic will deal with problems about why.

A mix in equal proportions in both orientations is said to be 'racémique' / racemate.
( This comes from Louis Pasteur study of acide racémique which contain both enantiomers
and so does not deflect polarized light)

A parting shoot to fix notions in memories :
Does one can say that "Z" and "S" laid cordages are cordages enantiomers!

Beware ANY knots can be, by altering some of its crossings ( not only a geometric
transformation but a knotting transformation too!) , made into a 'null knot' , that could
be quite inconvenient to say the least if you make inadvertently such a change..Be attentive.