Transcript The Unknot, the Trefoil Knot, and the Figure

The Unknot, the Trefoil Knot,
and the Figure-Eight Knot are
Mutually Nonequivalent
An Elementary Proof
Jimmy Gillan
Thursday, April 10, 2008
Outline
• Introduction
– A Brief History of Knot Theory
– What is a knot?
– Knot Terminology
• Defining Knots
• Equivalence and Knot Invariants
• The Proof
A Brief History of Knot Theory
• Carl Friedrich Gauss (1777 – 1855)
• Lord Kelvin, atoms, and the “ether”
• P.G. Tait first to formally publish a paper
on knots in 1877
– Enumerating and tabulating knots
• Early 20th century classical knot theory
becomes a formal branch of mathematics
• Knot theory now includes elements of
algebra, combinatorics and geometry
What is and what is not a knot?
•
The “string analogy”
•
Any tame mathematical knot can be
physically represented by the “string
analogy”
What is and what is not a knot?
Which of these two figures is a knot?
What is and what is not a knot?
Which of these two figures is a knot?
A knot!
Not a knot!
Some Terminology
• A knot diagram is defined as a pictorial
representation of a knot in R2
• Each diagram of a given knot K is defined
as a projection of the knot K.
Two knot diagrams
&
Two projections
of the same knot
Some Terminology
• A crossing is defined as a point in the
projection of a knot where the knot
intersects or crosses-over itself.
Defining Knots
Topological Definition
• A knot is an embedding of S1 in R3 or S3.
Simpler Definition
• A knot is a defined as a simple, closed
curve in R3 that is isotopic to a simple,
closed polygonal curve with a finite
vertex set.
Defining Knots
•
•
•
Let K be a curve in R3 and let f: I → R3 be
a continuous function such that f (I) = K.
closed – f (0) = f (1)
simple – if f (x) = f (y), then either x = y
or x,y ε {0,1}
Defining Knots
•
•
•
A simple, closed polygonal curve is defined
as follows:
Let (p1,…,pn) be an ordered set of points in R3
such that no three points lie on a common line
Let [pi, pj] denote the line segment between
points pi and pj
 n1

 [ pi , pi 1 ]  [ pn , p1 ]
 i 1

Defining Knots
•
Two curves are said to be isotopic if one
can be deformed to form the other in R3
without breaking the curve at any point
The simple, closed polygonal curve determined
by (a,b,c) and a knot to which it is isotopic.
Equivalence and Invariants
• How do we know if two different knot
diagrams represent different knots?
• Consider the two diagrams below, are
these projections of the same knot?
?
Equivalence and Invariants
• How do we know if two different knot
diagrams represent different knots?
• Consider the two diagrams below, are
these projections of the same knot?
√
YES!
Equivalence and Invariants
• Two projections are equivalent if you can
deform one into the other without breaking
the knot (“string analogy”)
• Planar isotopies –deformations that do not
change the crossings of a projection
Equivalence and Invariants
• Reidemeister moves
TYPE I
OR
TYPE II
OR
TYPE III
OR
Equivalence and Invariants
Alexander and Briggs Theorem
If two knot projections are equivalent, then their
diagrams are related by a series of Reidemeister
moves
• A knot invariant is defined as a characteristic of
a knot which is true for all of its projections
• Use knot invariants to determine whether two
knots are not equivalent
The Proof
INTRODUCTION
• Consider the three knots with the fewest
crossings in their simplest projections
– The unknot, O (0 crossings)
– The trefoil knot, T (3 crossings)
– The figure-eight knot, F (4 crossings)
The Proof
OUTLINE
O, T and F are mutually nonequivalent
Sketch of Proof
• First show T is not equivalent to O or F
using the knot invariant Tricolorability
• Then show O and F are not equivalent
using the knot invariant the Jones
Polynomial
The Proof
TRICOLORABILITY
• A knot is tricolorable if the pieces of a
projection that are not intersected in its diagram
can be colored with exactly 3 different colors
such that at each crossing in the knot is the
meeting of either 3 different colors or the
same color.
Is the trefoil knot T tricolorable?
The Proof
TRICOLORABILITY
• A knot is tricolorable if the pieces of a
projection that are not intersected in its diagram
can be colored with exactly 3 different colors
such that at each crossing in the knot is the
meeting of either 3 different colors or the
same color.
Is the trefoil knot T tricolorable?
YES!
The Proof
TRICOLORABILITY
• Is the unknot O tricolorable? NO!
• Is the figure-eight knot F tricolorable? NO!
The Proof
THE JONES POLYNOMIAL
• Assign a Laurent polynomial to each knot
• If two projections have different polynomials,
they are not equivalent
• J.W. Alexander [1928] - developed first knot
polynomial using matrices & determinants
• John Conway [1969] - calculate the Alexander
polynomial using skein relations
• Vaughn Jones [1984] developed a way to
calculate the Alexander polynomial using the
bracket polynomial and skein relations
The Proof
THE JONES POLYNOMIAL
• The Jones polynomial of a knot K is defined as:
3  w( K )
X ( K )  ( A )
K
where:
–
–
–
–
X(K) is the Jones polynomial of K
< K > denotes the bracket polynomial of K
w(K) denotes the writhe of K
A is the variable of bracket polynomial and A = t -1/4
The Proof
THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL
• There are three rules for computing the bracket
polynomial of a knot K
Rule 1 :
   1
Rule 2 :
 C   A  CV   A1  CH 
Rule 3 :
 K     (  A 2  A 2 )  K 
The Proof
THE JONES POLYNOMIAL – BRACKET POLYNOMIAL
• By Rule 1, <O> = 1
• Computing the bracket polynomial of F is more
involved
Take the projection of F and
enumerate the crossings
The Proof
THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL
• Consider crossing 1
• By Rule 2, <F> = A<FV> + A-1<FH> where FV and
FH are derivative knots created by changing
crossing 1 from C to CV and CH respectively
The Proof
THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL
• We must use Rule 2 again with crossing 2 in
order to compute both <FV> and <FH>, giving us
four derivative knots, <FVV>, <FVH>, <FHV>, and
<FHH> with:
 FV   A  FVV   A1  FVH 
and  FH   A  FHV   A1  FHH 
• By substituting into the formula for <F> we get:
 F   A( A  FVV   A1  FVH  )
 A1 ( A  FHV   A1  FHH )
The Proof
THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL
• Continue expansion with Rule 2 until all
crossings have been eliminated
Clearly FVHHV is
planar isotopic to
the unknot and
since <O> = 1 by
Rule 1, <FVHHV> = 1
The Proof
THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL
• Now Rules 1 and 3 can be used to compute the
values of the resulting 16 derivative knots and
have <F> in terms of A
• After lots of drawing and simplification we
ultimately get:
4
8
 F   A  A 1 A  A
8
4
The Proof
THE JONES POLYNOMIAL – WRITHE
• The writhe of F, w(F),
is computed as
follows:
The Proof
THE JONES POLYNOMIAL – WRITHE
• The writhe of F, w(F),
is computed as
follows:
– Give F an orientation
The Proof
THE JONES POLYNOMIAL – WRITHE
• The writhe of F, w(F),
is computed as
follows:
– Give F an orientation
– Assign +1 or -1 to the
crossings according
to its type
+ 1 crossing
– 1 crossing
The Proof
THE JONES POLYNOMIAL – WRITHE
• The writhe of F, w(F),
is computed as
follows:
– Give F an orientation
– Assign +1 or -1 to the
crossings according to
its type
– Sum the assignments
over all crossings
w( F )  1  1  1  1  0
The Proof
THE JONES POLYNOMIAL – COMBINING THE TWO
• Substitute the bracket polynomial and writhe of
F and O into the original equation and replace A
with t -1/4
4
8
X ( F )  ( A ) ( A  A  1  A  A )
3 0
t
2
1
8
4
 t 1 t  t
2
X (O)  ( A3 ) 0 (1)  1
• Clearly X(F) ≠ X(O)
The Proof
CONCLUSION
• T is not equivalent to O and T is not
equivalent to F because T is tricolorable
and O and F are not
• O and F are not equivalent because their
Jones polynomials are different
• Thus O, T and F are mutually
nonequivalent
□
Thank You
I’d like to thank my advisors, Professor
Ramin Naimi and Professor Ron Buckmire,
and the Occidental Mathematics
Department for all their help and support
over the last four years.
Thank you for coming!
References
• Adams, Colin C., The Knot Book: An Elementary
Introduction to the Mathematical Theory of Knots,
(New York, NY: W.H. Freeman and Company,
1994)
• Kauffman, Louis H., On Knots, (Princeton, NJ:
Princeton University Press, 1987)
• Livingston, Charles, Knot Theory, (Washington,
D.C.: Mathematical Association of America, 1993)