The Unknot, the Trefoil Knot, and the Figure-Eight Knot are Mutually Nonequivalent An Elementary Proof

The Unknot, the Trefoil Knot, and the Figure-Eight Knot are

Mutually NonequivalentAn Elementary Proof

Jimmy GillanThursday, 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”


Which of these two figures is a knot?


A knot! Not a knot!

Which of these two figures is 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 projections of the same knot

Two knot diagrams

&

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

],[],[ 1

1

11 pppp n

n

iii

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?

?


• 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!


• 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


• Reidemeister moves

OR

OR

OR

TYPE I

TYPE II

TYPE III


Alexander and Briggs TheoremIf 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

• 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)

INTRODUCTION

The Proof

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

O, T and F are mutually nonequivalent

OUTLINE

The Proof

• 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?

TRICOLORABILITY

The Proof

• 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!

TRICOLORABILITY

The Proof

• Is the unknot O tricolorable? NO!

• Is the figure-eight knot F tricolorable? NO!

TRICOLORABILITY

The Proof• 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 JONES POLYNOMIAL

The Proof

• The Jones polynomial of a knot K is defined as:

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

KAKX Kw )(3)()(

THE JONES POLYNOMIAL

The Proof

• There are three rules for computing the bracket polynomial of a knot K

KAAK

CACAC HV

)( :3 Rule

:2 Rule

1 :1 Rule

22

1

THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL

The Proof

• By Rule 1, <O> = 1• Computing the bracket polynomial of F is more

involved

Take the projection of F and enumerate the crossings

THE JONES POLYNOMIAL – BRACKET POLYNOMIAL

The Proof

• 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

• 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:

• By substituting into the formula for <F> we get:

HHHVH

VHVVV

FAFAF

FAFAF1

1

and

)(

) (11

1

HHHV

VHVV

FAFAA

FAFAAF


The Proof

• 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

• 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:

8448 1 AAAAF


The Proof

• The writhe of F, w(F), is computed as follows:

THE JONES POLYNOMIAL – WRITHE

The Proof

• The writhe of F, w(F), is computed as follows:– Give F an orientation


The Proof

• 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 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


01111)( Fw

The Proof• Substitute the bracket polynomial and writhe of F and

O into the original equation and replace A with t -1/4

• Clearly X(F) ≠ X(O)

THE JONES POLYNOMIAL – COMBINING THE TWO

212

844803

1

)1()()(

tttt

AAAAAFX

1)1()()( 03 AOX

The ProofCONCLUSION

• 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)

Documents

The Unknot, the Trefoil Knot, and the Figure-Eight Knot are Mutually Nonequivalent An Elementary Proof