Knots and their

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Knots and their mirror images
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Question Are the two knots ambient isotopic?
Answer No. There is no such an ambient isotopy
that would bring the knots to the same
conformation. (Tie the knots and try to find the
isotopy!) Conclusion There are two different
knots (knot types) known under the same name
trefoil knot.
31
31
The trefoil knot and its mirror image.
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Question Are the two knots ambient isotopic?
Answer Yes. There exists an ambient isotopy that
brings the two knots to the same shape.(We shall
demonstrate it later.) Conclusion There is only
one knot (knot type) known under the name figure
eight knot.
41
41
Figure eight knot and its mirror image
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41 knot is amphichiral. Here is the ambient
isotopy which transforms it into its mirror image
K
K
planar isotopy
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The problem of composite knots
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Question (stupid?) Is a connected sum of two
prime knots always of the same knot
type? EXPERIMENT Let K1 and K2 be two
non-trivial prime knots. For instance K1
31 K2 52
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...let us connect the knots into a single knot...
Question Will the knot type be the same, if the
connection between the K1 and K2 knots is made
at different locations ?
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Connections between the knots at each of the
indicated locations give composite knots of the
same knot type there is only one 3152 knot.
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Is it always like that? NO!
The composite knots shown above are both 817
817 but they are of different knot type!
Question How are the compositions made and
what is it what makes them different?
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Let us take a pair of the 817 knots ...
... and let us rotate the right knot by ? around
the marked axis...
Knots within each pair can be connected giving
two composite knots...
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Are the knots equivalent?
(817817)A
For the sake of convenience we denote them
differently
Both knots are 817817 .
(817817)B
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is there such an ambient isotopy ?
(817817)A
Had we been able, using an ambient isotopy, to
convert the lower knot into the upper knot we
would have proven their equivalence.
(817817)B
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To prove that the knots cannot be converted via
an ambient isotopy into each other let us first
orient them ...
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(817817)A
(817817)B
Factor knots on the left are congruent and they
are oriented in the same direction.
But the factor knots on the right are not
congruent.
A rotation by ? around the indicated axis makes
the factor knots congruent but their orientations
are now opposite.
The left factor knots are congruent and they are
oriented in the same direction.
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The lower factor knot can be made congruent with
the upper factor knot by a rotation.
Unfortunately, it changes orientation of the
knot. This is not allowed since such a change
would have been never observed had the knot been
all time connected to the left factor knot.
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We can imagine that for the time of all ambient
isotopic manipulations with the right factor knot
the size of the left factor knot is strongly
reduced and that its conformation becomes frozen.
At the end of the manipulations the original size
and position in space is recovered. Obviously,
since the conformation of the knot remained
frozen all time, its orientation remained intact.
Orientation of the right factor knot must fit it.
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NO AMBIENT ISOTOPY CAN DO IT
The 817 knot is particular if we orient it,
there is no ambient isotopy that would invert the
orientation the knot is not invertible.
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Terminology
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CLASSIFICATION OF ORIENTED KNOTS Let K be an
oriented knot. Let K denotes the mirror image of
K. Let -K denotes the knot obtained from K by
inverting its orientation. 1. K -K invertible
or reversible. 2. K K amphichiral (not
amphichiral chiral) 3. K -K involutive. REM
ARK. (after de la Harpe) Some writers use
"amphichiral" to mean "amphichiral or
involutive".
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CLASSIFICATION OF NON-ORIENTED KNOTS Let K be a
knot, not oriented. Let K denotes the mirror
image of K. 1. K K amphichiral ATTENTION
! If the unoriented knot classified as
amphichiral becomes oriented we may find out
that it is equivalent to the mirror image with
an inverted orientation i.e. within the
classification of oriented knots it is not
amphichiral but involutive.
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EXAMPLES 1. The 31 knot is invertible but not
amphichiral, thus, it is not involutive. 2.
The 41 knot is invertible and amphichiral,
thus, it is involutive. 3. The 817 knot is not
invertible and not amphichiral, but it is
involutive. 4. The 932 knot is not invertible,
not involutive and not invertible. Thus, 932 ?
- 932 ? 932 ? - 932. There are 4 different
kinds of the oriented 932 knot. 5. The 932
932 knot is amphichiral and not involutive,
thus, it is not invertible.
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The 31 knot is invertible.
?
Here is a particular conformation at which the
orientation of the knot can be inverted just by
a rotation.
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The 31 knot is not amphichiral.
31 knot its mirror image
There is no ambient isotopy, in particular - no
rotation, that would transform the knot on the
left into the knot on the right
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The 41 knot is invertible
?
The oriented 41 knot and ... its inverted
version.
In this particular conformation of the knot its
orientation can be inverted by a simple rotation.
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The 41 knot is amphichiral and invertible,
thus, it is involutive
K
-K
K
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ROTATION BY ?/2
41
41
MIRROR REFLECTION
41
A particular conformation of the 41 knot for
which a mirror reflection can be achieved by a
simple rotation