1 Knot Theory 2 Tricolorability Knot Theory A mathematicians knot formally speaking is a closed loop in R 3 I t is a line that we can draw in the space ID: 547124
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Slide1
The Miracle of Knot
1
.
Knot Theory
2
.
TricolorabilitySlide2
Knot Theory
A
mathematician's
knot,
formally
speaking,
is
a
“closed loop”
in
R
3
.
I
t
is a
line
that
we can draw in the space
R
3
which
does not intersect itself and
go
back to where
it started
.
Two
mathematical knots are
considered the same
if one can be
“bended”
into the other
knot
in
R
3
.
The way of bending should
not involve cutting the
line
or passing the
line
through itself
.Slide3
Different KnotsSlide4
UnknotThis is important to know that the simplest knot is the unknot, because people often want to figure out about when a knot is the unknot
.
In the definition, we talked about knots in
R
3
, however people usually work on knots in
R
2
, such as what has been shown in previous pictures.
It’s because working on knots in
R
3
is much more difficult, and we do it as how we draw a cube in
R
2
.
S
olid lines demonstrate they are in the front
, and
lines that are “cut” are in the back. Slide5
Is a Given Knot the Unknot?Slide6
Reidemeister
moves
R
eidemeister
moves
are a set of ways that we can re-draw parts of a knot in
R
2
by not changing the knot.
(1) Twist
and untwist in either direction.
(2) Move
one
line
completely over another.
(3) Move
a
line
completely over or under a crossing
.Slide7
Reidemeister movesSlide8
Reidemeister
moves on a knot
.Slide9
One of the Ways to Define a Knot
Tricolorability
In
the mathematical field of knot theory, the
tricolorability
of
a knot
is the ability of a knot to be colored with three colors
according
to certain rules.
Tricolorability
will not be changed when we draw a knot in other
ways by using
Reidemeister
moves.
In this case, we can know that if one knot can be tricolored and the other cannot,
then they
must be different knots.Slide10
Rules of tricolorabilityA knot is tricolorable
if each
line
of the knot diagram can be colored one of three colors, subject to the following rules
:
(1)At
least two colors must be
used
(2)At
each crossing, the
three lines that leave the crossing
are either all the same color or all different colors.Slide11
Tricolorable
or not
YES NoSlide12
Reidemeister
Moves
Are
T
ricolorable
.
Twist to Untwist
Unpoke
to poke
SlideSlide13
Conclusion
A
tricolorable
knot can’t transfer to an un-
tricolorable
knot by using
reidemeister
moves. The opposite way can’t either.
By using this property, we can know if a given complex knot
is
tricolorable
, then it must not be the unknot.
The
unknot is not
tricolorable
, any
tricolorable
knot is necessarily nontrivial
.Slide14
Refhttp://en.wikipedia.org/wiki/Knot_theory
http://
en.wikipedia.org/wiki/Tricolorability
http://mit.uvt.rnu.tn/NR/rdonlyres/Mathematics/18-304Spring-2006/94568A74-E63B-4C2C-82C2-01C14CFD9CF8/0/jacobs_knots.pdf