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Knot Theory

Definition
:
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1)
A knot is
a simple closed curve in 3dimensional space.
What
does that mean?
Well, a loop like the one at the left is considered
a knot in mathematical knot theory (it is a simple closed curve in
3dimensional space). In fact this knot has a special name:
The unknot. The
unknot can be
drawn with no crossings, and is also called a
trivial knot.
It is the simplest of all knots.
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2)
The Central Problem of Knot Theory
The central problem of Knot Theory is determining whether two
knots can be rearranged (without cutting) to be exactly alike.
A
special case of this problem is one of the fundamental questions
of Knot Theory: Given
a knot, is it the unknot? Now, for a simple
loop, that's an easy question. But, take for example the trefoil
knot animated at the top of the page. Is it possible to transform
this knot so that it looks like the unknot? Tie a trefoil knot
yourself and see if you can untangle it to form a simple circular
loop.
When we actually start trying to untangle and rearrange knots to
look like one another, we begin what can seem like a very
complicated process. Mathematicians were perplexed at the
seemingly unending number of ways a knot could be shaped and
turned. What was needed was a simple set of rules for working with
knots.
Finally, German mathematician Kurt Reidemeister (18931971) proved
that all the different transformations on knots could be described
in terms of three simple moves. The next section will give us the
simple tools we need to begin working with knots in a mathematical
context.
3)
How do we work with knots? (The Reidemeister moves)
In
1926, Kurt Reidemeister
(rideamystir) proved that if we have different
representations (or
projections)
of the same knot, we can get one to look like the other using just
three simple types of moves.
The Reidemeister Moves 
1. Take out
(or put in) a simple twist in the knot:


2. Add or
remove two crossings (lay one strand over another):


3. Slide a strand from one side of a crossing to the other:


These three types of moves are called
Reidemeister
moves.
The
Activities Page
contains knot activities that have you use these three moves to
find out if two knots are indeed equivalent.
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4)
Classifying different knots
One of the first objectives mathematicians wished to achieve in
working with knots was formulating tables of distinct knots. To be
able to classify knots, it is easiest if we work with only one
projection (or representation) of each knot to avoid duplication.
First, we must "simplify" the knot as much as possible. This means
we use the Reidemeister moves (from above) to get as few crossings
in the knot as possible. Once we simplify the knot so that we
cannot remove any further crossings, the knot is classified by the
number of crossings that remain. For example, the trefoil knot is
classified by its fewest number of crossings  three (see the
diagram below).
Sometimes it is possible to have more than one knot with the same
number of crossings. In this case, we usually use subscripts to
denote different knots with the same number of crossings, such as
the 5_{1}
and 5_{2}
knots in the diagram below:
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5)
Properties of knots
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We can change the way a knot looks so much that it
can be hard to tell what we started with.
So, what stays the same about a knot in different projections?
Knots have some properties that depend only on the knot itself and
not on how it is looking at any particular moment. These
properties are called
invariants
of the knot.
One invariant is the
minimal crossing
number.
The minimal
crossing number of a knot is the least number of crossings that
appear in any projection of the knot. For example, the
unknot has a minimal crossing number of 0. The trefoil knot has a
minimal crossing number of 3. The classifications in number 4
above rely on this minimal crossing number. No matter how much we
tangle a knot (without cutting), it can always be simplified to
its minimal number of crossings using the Reidemeister moves.
Another invariant is the
unknotting number.
The unknotting number is the least
number of crossing changes necessary to turn a knot into the
unknot. By "crossing changes" we mean changing the
orientation of two strings where they cross. The trefoil knot can
be unknotted by changing only one crossing from under to over, so
the trefoil knot has an unknotting number of 1 (try this to verify
it).
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6)
Close cousins  Knots vs. Links
Is the figure to the right a
knot?
Sometimes we run into figures that cannot be classified as knots.
This figure at the right looks kind of like a knot, but we no
longer have a simple
closed curve  we have a
group of simple closed curves (three separate loops).
The
new figure is called a link.
A link
is a collection of knots; the individual knots which make up a
link are called the
components of the link.
This specific link shown above is known as the
Borromean Rings.
An interesting fact about the Borromean Rings:
If you remove one of
the component loops, the other two loops will no longer be
connected!
An interesting
question:
Can
the Borromean Rings be formed using 3 flat closed loops? Not sure?
Give it a try!
Just as mathematicians try to untangle knots to form the unknot,
we try to separate links to form the "unlink". A link is referred
to as
splittable if the component loops can be separated without cutting...
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Learning more about knots :
Knot Theory