Transcript The Kauffman Bracket in Knot Theory

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To show that the Kauffman Bracket is
unchanged under each of the three
Reidemeister moves.
First explain the basics of knot theory.
Then show you what the Reidemeister moves
are, and how they affect knots.
Finally, illustrate the Kauffman Bracket and it’s
applications.
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Given a loop, is it really knotted, or can it
be untangled without having to be cut?
Given two Knots how do you tell
whether the two Knots are the same or
different?
Is there an effective algorithm to answer
the above questions?
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A piece of string with a knot tied in it
Glue the ends together because it is
necessary in knot theory that a knot be a
continuous loop.
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If you deform a knot in the plane without
cutting it, it doesn’t change.
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Goes under, over, under, over…
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The simplest knot.
An unknotted circle, or the trivial knot.
You can move from the one view of a
knot to another view using a series of
Reidemeister moves.
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Reidemeister moves change the
projection of the knot.
This in turn, changes the
relation between crossings, but
does not change the knot.
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First: Allows us to put
in/take out a twist.
Second: Allows us to
either add two
crossings or remove
two crossings.
Third: Allows us to
slide a strand of the
knot from one side of a
crossing to the other.
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Given two knots K1 and K 2 , then K1  K2 iff
you can get from K1 to K 2 by a series of
Reidemeister Moves.
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A set of disjoint knots.
The classic Hopf Links
with two components
and 10 components.
The Borremean Rings
with three
components.
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The linking number is a way of measuring numerically
how linked up two components are.
If there are more than two components, add up the
link numbers and divide by two.
Positive crossing +1
If you rotate the under-strand
clockwise they line up
Negative crossing -1
If you rotate the under-strand
counterclockwise they line up
+1
+1
-1
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+1
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Rotate under strand
Linking number is
1111
2
2
+1
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The linking number is unaffected by all
three Reidemeister moves. Therefore it is
an invariant of the oriented link.
+1
-1
-1
+1
-1
+1
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To calculate the writhe you must start with an
oriented knot or link projection.
At each crossing you will have either a +1 or a -1.
The writhe is the sum of these numbers.
It is expressed as  ( L) where L is a link projection.
1+1-1-1=0
+1
-1
So the writhe of the
figure eight knot is 0
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The writhe of a knot projection is
unaffected by Reidemeister moves II and III
thus becomes useful for other knot
polynomials, such as the Kauffman X
polynomial.
+1
-1
+1
-1
-1
+1
+1
+1
-1
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Three Rules
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In order to show that a given
polynomial is in fact knot/link invariant,
it is necessary and sufficient to show
that the invariant in question is
unchanged under each of the three
Reidemeister moves.
I.E. I need to show that the Kauffman
Bracket is unchanged under each of the
three Reidemeister moves.
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We want the polynomials to satisfy the
Reidemiester moves so, let’s find B in terms of A.
The same is done to determine C:
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Three Rules
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This shows that the bracket polynomial is
invariant under Reidemeister moves II
and III i.e. is an invariant of regular
isotopy.
It is not an invariant under Reidemeister I
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The next task is to find a way to make it work.
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Planar Isotopy is the motion of a diagram in
the plane that preserves the graphical
structure of the underlying projection.
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A knot or a link is said to be Ambient isotopic
to another if there is a sequence of
Reidemeister moves and planar equivalences
between them.
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In a type 1 Reidemeister move the
Kauffman Polynomial looks like
X  L     A3 
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We know
 ( L )
L
  A2  A2
Kauffman
Bracket
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Since the writhe of the unknot is 0, then
X
  A
2
2
A
Kauffman X
Polynomial
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So, now the Type I move can be calculated:
Now, X(L) is invariant since it does not
change with any of the Reidemeister moves.
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Hopf Link
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Labeling a crossing
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Labeling a Projection
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Crossing
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A-Split
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B-Split
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The choice of how to split all n-crossings in a
projection L.
 Denoted:
S
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1 2
Notice that this particular state of the trefoil contributes A A
to the bracket polynomial of this projection
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The bracket polynomial of a link L will
now simply be the sum over all the
possible states.
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Denoted:
L  S A
a(S )
b ( S )
A
A
2

2 S 1
A
Where a  S  is the number of A-splits in S, and b  S 
is the number of B-splits in S.
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To compute the bracket polynomial of the
trefoil knot.
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There are 3 crossings so there will be 23  8 states
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It is hard to tell the unknot from a messy
projection of it, or for that matter, any knot
from a messy projection of it.
If L does not equal
same knot as L2.
1
A7  A3  A5
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L2
, then L1 can’t be the
A7  A3  A5
However, the converse is not necessarily true.
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I have proved that
the Bracket
polynomial is
invariant under the
three Reidemeister
moves.
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Knot Theory has many applications
in other fields, specifically biology.
Knot theory aids in DNA research of
knotted or tangled DNA.
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Pictures taken from
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http://www.cs.ubc.ca/nest/imager/contribu
tions/scharein/KnotPlot.html
Other information from
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The Knot Book, Colin Adams
Complexity: Knots, Colourings and
Counting, D. J. A. Welsh
New Invariants in the Theory of Knots, Louis
Kauffman
Jo Ellis-Monaghan
Labeling Technique for Shaded
Knot/Link Projections
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Begin with the shaded knot projection.
If the top strand ‘spins’ left to sweep out black then it’s a +
crossing.
If the top strand ‘spins’ right then it’s a – crossing.
+
-
Links to Planar Graphs
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Given a knot projection
Shade it so that the outside region is
white/blank.
Put a vertex inside each shaded region.
Connect the regions if the shaded areas
share a crossing.
Kauffman Bracket In
Polynomial Terms
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if e is an edge corresponding to:
A negative crossing:
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There exists a graph G such that G  A G  e  A1 G / e
where G  e and G / e denote deletion and contraction
of the edge e from G
A positive crossing:
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There exists a graph G such that G  A G / e  A G  e