Knot Theory for Schools (Dec 07)

8/2/2019 Knot Theory for Schools (Dec 07)

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What Is a Knot

When Are Two Knots the Same

Tricolourability

Supercoiling

Knots and Telephone WiresA Light Introduction to Knot Theory

Tony Skyner

Department of MathematicsUniversity of Bristol

Maths for Schools 2007

Knot Theory


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What Is a Knot


Tricolourability

Supercoiling

Outline

1 What Is a KnotDefinition of a Knot

Examples of knots and Topology

A Brief History of Knots2 When Are Two Knots the Same

Reidemeister MovesExample

3 Tricolourability

Knot Colouring

Reidemeister MovesExercise

Knot Arithmetic4 Supercoiling

Ribbons

SupercoilingKnot Theory


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What Is a Knot


Tricolourability

Supercoiling

Outline





3 Tricolourability

Knot Colouring

Reidemeister Moves

Exercise


Ribbons



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What Is a Knot


Tricolourability

Supercoiling

Outline





3 Tricolourability

Knot Colouring

Reidemeister Moves

Exercise


Ribbons



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What Is a Knot


Tricolourability

Supercoiling

Outline





3 Tricolourability

Knot Colouring

Reidemeister Moves

Exercise


Ribbons


Wh I K


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What Is a Knot


Tricolourability

Supercoiling

Definition of a Knot


A Brief History of Knots

Defining a Knot

A Knot is:

A closed loop of string.

Made of very flexible rubber.Otherwise obeying the laws ofphysics:

We cannot cut the loop.We cannot pass one bit

through another.We cannot pull the knot sotight it disappears.

Knot Theory

Wh t I K t


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a Knot


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a Knot


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a Knot


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a Knot


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a KnotD fi iti f K t


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What Is a Knot


Tricolourability

Supercoiling




Defining a Knot

A Knot is:





Knot Theory

What Is a KnotD fi iti f K t


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Tricolourability

Supercoiling




What Can we do with a knot

We can stretch and squeeze aknot.

We can twist or rotate a knot.

We can push different parts of

the knot around.

We call these deformations

Knot Theory

What Is a KnotDefinition of a Knot


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Tricolourability

Supercoiling








the knot around.


Knot Theory



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Tricolourability

Supercoiling








the knot around.


Knot Theory



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Tricolourability

Supercoiling








the knot around.


Knot Theory

What Is a Knot

SDefinition of a Knot


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Tricolourability

Supercoiling




The Same Knot, Only Different

We consider two knots the same if we can get from one to the

other by deforming it in any of the ways stated before.

Knot Theory

What Is a Knot

Wh A T K t th SDefinition of a Knot


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Tricolourability

Supercoiling

e t o o a ot



Drawing Knots

We draw knots as

projections.

Imagin light is shining on the

knot, making a shade.Points where one bit of the

knot goes under another is

called a crossing.

Crossings and shown by small

gaps in the knot.

We call a piece of knot

between two under-crossings

a strand.

Knot Theory

What Is a Knot

When Are Two Knots the SameDefinition of a Knot


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Tricolourability

Supercoiling



Drawing Knots

We draw knots as

projections.


knot, making a shade.

Points where one bit of the


called a crossing.


gaps in the knot.



a strand.

Knot Theory

What Is a Knot



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Tricolourability

Supercoiling



Drawing Knots

We draw knots as

projections.





called a crossing.


gaps in the knot.



a strand.

Knot Theory

What Is a Knot



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Tricolourability

Supercoiling



Drawing Knots

We draw knots as

projections.





called a crossing.


gaps in the knot.



a strand.

Knot Theory

What Is a KnotWhen Are Two Knots the Same



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Tricolourability

Supercoiling



Drawing Knots

We draw knots as

projections.





called a crossing.


gaps in the knot.



a strand.

Knot Theory



E l f k d T l


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Tricolourability

Supercoiling



Example

For example:

Knot Theory



E l f k t d T l


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Tricolourability

Supercoiling



The Unknot

The simplest knot has no

crossings.

We call it the Unknot.

Knot Theory





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Tricolourability

Supercoiling



The Unknot

The simplest knot has no

crossings.

We call it the Unknot.

Knot Theory





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Tricolourability

Supercoiling



Topology

Knot Theory is a branch of a

big subject, called Topology.

Topologist study geometric

objects (such as squares,cubes, spheres etc).

They consider objects that

differ only by deformation to

be the same.

So, for example, they consider

a cube and a sphere to be the

same.

Knot Theory





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Tricolourability

Supercoiling



Topology







be the same.



same.

Knot Theory





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Tricolourability

Supercoiling



Topology







be the same.



same.

Knot Theory


T i l bilit




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Tricolourability

Supercoiling



Topology







be the same.



same.

Knot Theory


Tricolourability




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Tricolourability

Supercoiling

p p gy


Topology







be the same.



same.

Knot Theory


Tricolourability




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Tricolourability

SupercoilingA Brief History of Knots

Topology







be the same.



same.

Knot Theory


Tricolourability




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Tricolourability


Topology







be the same.



same.

Knot Theory


Tricolourability




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Tricolourability


Topology







be the same.



same.

Knot Theory


Tricolourability



A B i f Hi f K


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Tricolourability


The Trefoil

The next simplest knot, afterthe unknot, is the Trefoil Knot.

We usually draw it, like in the

diagram (right).

Knot Theory


Tricolourability



A B i f Hi t f K t


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y


The Trefoil

The next simplest knot, afterthe unknot, is the Trefoil Knot.

We usually draw it, like in the

diagram (right).

Knot Theory


Tricolourability





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y


A different Trefoil

But what about this knot?

It is clearly the trefoil, and we

could easily remove the extra

twist.

Knot Theory


Tricolourability





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A different Trefoil

But what about this knot?

It is clearly the trefoil, and we

could easily remove the extra

twist.

Knot Theory


Tricolourability





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The Trefoil???

How about this knot?Is it the trefoil?

In fact, it is, but it is difficult to

see how we need to deform it.

Knot Theory


Tricolourability





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The Trefoil???




Knot Theory


Tricolourability

S ili





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The Trefoil???




Knot Theory


Tricolourability

S ili





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Supercoilingy

Are there knots

Are there actually knots, or are

they all deformed versions of

the unknot.

How do we tell when two knots

are the same, or different.

We could try and deform one

to the other, but how long to

we keep going before givingup?

Knot Theory


Tricolourability

Supercoiling





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Supercoilingy

Are there knots



the unknot.






Knot Theory


Tricolourability

Supercoiling





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Supercoiling

Are there knots



the unknot.






Knot Theory


Tricolourability

Supercoiling





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Supercoiling

Why study knots

Time for some history.

As much as knots appear to be an interesting in their own right,

why would we want to study them.

Knot Theory


Tricolourability

Supercoiling





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Supercoiling

Lord Kelvin and the Atom

In the 1880s, it was believed

that a substance called ether

pervaded all of space.

Lord Kelvin hypothesized thatatoms were knots in the ether.

So different knots

corresponded to different

atoms.This convinced Peter Tait to

compile a list of distinct knots.

Knot Theory


Tricolourability

Supercoiling

Definition of a KnotExamples of knots and Topology



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Supe co g






So different knots




Knot Theory


Tricolourability

Supercoiling




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p g






So different knots




Knot Theory


Tricolourability

Supercoiling




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So different knots




Knot Theory


Tricolourability

Supercoiling




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Chemists, Biochemists and Physisist

Although Lord Kelvin waswrong, the study of knots

continued for over 100 years.

Biochemists have found

knotted structures in DNA.Chemists have found they canknot molecules.

and the properties of themolecule are determineed

by the type of knot!Now, knot theory is important

in Chemistry, Biology and

Physics, as well as

mathematics in its own right.

Knot Theory


Tricolourability

Supercoiling




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Physics, as well as


Knot Theory


Tricolourability

Supercoiling




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Physics, as well as


Knot Theory


Tricolourability

Supercoiling




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Physics, as well as


Knot Theory


Tricolourability

Supercoiling




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Physics, as well as


Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example


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How to deform a knot

Instead of just playing with bits of string, we shall only work

with the projections.

We would like a set of rules for deforming a knot.

The first is called ambient isotopy.

Dont worry about the words - it just means we can stretchand squish the knot.But the knot has to look sort of the same

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

H d f k


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Th R id i t M


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The Reidemeister Moves

In fact, there are only three more rules.They are called The Reidemeister Moves.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Th R id i t M


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The Reidemeister Moves

In fact, there are only three more rules.They are called The Reidemeister Moves.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

R id i t M I


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Reidemeister Move I

This move either adds orremoves a twist.

It either creates or removes

one crossing.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Reidemeister Move I


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Reidemeister Move I

This move either adds orremoves a twist.


one crossing.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Reidemeister Move II


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This move passes one strand

under or over another.

Or pulls one strand out from

under, or over, another.


two crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example



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two crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example



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two crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Reidemeister Move III


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This one moves a peice of the

knot from one side of a

crossing to another.

Note that this is either above

or below the crossing.

You must be careful as to not

create or remove any

crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example



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crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example



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crossings.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Reidemeister Moves

Example

Example Time


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Example Time

Time for an example.

We are going to change the

knot, right.

Knot Theory

What Is a Knot

When Are Two Knots the SameTricolourability

Supercoiling


Example Time


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Example Time

Time for an example.

We are going to change the

knot, right.

Knot Theory

What Is a Knot


Supercoiling


Example Time


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Example Time

So that it looks like the knot,left, using the Reidemeister

Moves (and ambient

isotopies).

Knot Theory

What Is a Knot


Supercoiling


Example Time


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Example Time

The Knots look a little like this:

Knot Theory

What Is a Knot


Supercoiling


Step One/Six


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p

Use ambient isotopy.

Knot Theory

What Is a Knot


Supercoiling


Step Two/Six


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p

Use RMII (and ambient isotopy)

Knot Theory

What Is a Knot


Supercoiling


Step Three/Six


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p

Use RMII twice.

Knot Theory

What Is a Knot


Supercoiling


Step Four/Six


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p

Use RMIII.

Knot Theory

What Is a Knot


Supercoiling


Step Five/Six


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p

Use RMII (in reverse).

Knot Theory

What Is a Knot


Supercoiling


Step Six/Six


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Use RMI.

Knot Theory

What Is a Knot


Supercoiling


Kurt Reidemeister


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Kurk Reidemeister proved that

his moves and ambientisotopies are all you need to

deform one knot to another.

The proof is fairly simple, but

very long.

Knot Theory

What Is a Knot


Supercoiling


Kurt Reidemeister


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Kurk Reidemeister proved that

his moves and ambientisotopies are all you need to

deform one knot to another.

The proof is fairly simple, but

very long.

Knot Theory

What Is a Knot


Supercoiling


The Problem


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There is a problem with using the Reidemeister moves to

show that two knots are the same.

We could try 1,000,000 moves, without success and so

give up, but one more move would do.

We could try 1,000,000,000,000,000 moves, and one more

move would have done it!

We can only show (if we are lucky) that two knots are the

same, but not that they are different.

Knot Theory

What Is a Knot


Supercoiling


The Problem


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Knot Theory

What Is a Knot


Supercoiling


The Problem


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Knot Theory

What Is a Knot


Supercoiling


The Problem


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Knot Theory

What Is a Knot


Supercoiling


Invariants


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So what we want is some quantity that:

We can easily calculate it from the knot projection.

If two knots are the same (i.e. we can deform one to theother) then they have the same value.This value is unique for each knot.

Such a quantity is called an invariant.

Knot Theory

What Is a Knot


Supercoiling


Invariants


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Knot Theory

What Is a Knot


Supercoiling


Invariants


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Knot Theory What Is a Knot


Supercoiling


Invariants


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Supercoiling


Invariants


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Supercoiling


However...


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Unfortunately, it is vary hard to get invariants that are

unique.We have a number of invariants, but many knots share the

same value.



Supercoiling


However...


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Unfortunately, it is vary hard to get invariants that are

unique.We have a number of invariants, but many knots share the

same value.



Supercoiling

Knot Colouring


Knot Arithmetic


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We shall now consider one of the simplest invariants -

tricolourability.



Supercoiling

Knot Colouring


Knot Arithmetic

Tricolourability


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We say a knot is tricolourableif:

1 At each crossing, either allthe strands are the samecolour, or they are differentcolours.

2 At least two colours areused.



Supercoiling

Knot Colouring


Knot Arithmetic

Tricolourability


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Supercoiling

Knot Colouring


Knot Arithmetic

Tricolourability


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Supercoiling

Knot Colouring


Knot Arithmetic

Examples


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Lets try colouring some knots.

First, the Unknot.



Supercoiling

Knot Colouring


Knot Arithmetic

The Unknot


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The Unknot is not tricolourable,because we can only use one

colour.



Supercoiling

Knot Colouring


Knot Arithmetic

The Trefoil


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Next, The Trefoil.



Supercoiling

Knot Colouring


Knot Arithmetic

The Trefoil


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The Trefoil is tricolourable, wherewe colour each strand a different

colour.

Knot Theory

What Is a Knot


Supercoiling

Knot Colouring


Knot Arithmetic

The Figure-Eight


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Finally, lets try and colour the

Figure-Eight Knot.

Knot Theory

What Is a Knot


Supercoiling

Knot Colouring


Knot Arithmetic

The Figure-Eight


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Start by colouring any strand (itdoes not matter which one we

choose).

Knot Theory

What Is a Knot


Supercoiling

Knot Colouring


Knot Arithmetic

The Figure-Eight


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Look at the crossing at the

bottom.

We can either make that

crossing all red, or we havedifferent colours meeting.

If we choose all red, we run

into problems - the last strand

will also have to be red, and

we have not used at least 2colours.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The Figure-Eight


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bottom.







Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The Figure-Eight


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bottom.







Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The Figure-Eight


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So we shall colour the two

strands different colours.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The Figure-Eight


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Finally, to make the middle

crossing valid, we must use a

different colour.

But this leaves us with a

problem. So the Figure-Eight

is not tricolourable.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The Figure-Eight


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Finally, to make the middle

crossing valid, we must use a

different colour.

But this leaves us with a

problem. So the Figure-Eight

is not tricolourable.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Tricolourablility is an Invariant


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We need to prove that tricolourability is an invariant.

That is, if a knot is tricolourable, then any deformed versionof it is also tricolourable.

We can do this using the Reidemeister Moves.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Reidemeister Move I


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Suppose we have a coloured

knot, and part of it looks like

the picture on the left.

If we add a twist using

Reidemeister Move I, then we

can keep the two strands the

same colour - and this is valid.

As the two original strands are

the same colour as before, therest of the knot is unaffected.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Reidemeister Move I


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Reidemeister Move I


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Now suppose part of knot

looks like the left hand image.

Applying Reidemeister Move II

creates a new strand.

Regardless of the colour of the

original two strands, we can

set our new strand to be (one

of) the other colour(s).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Reidemeister Move III can be

shown to preserve

tricolourability in the same

way.

However, there a quite a few

more cases to consider, so

lets skip this (and just believe

me that it works!!!!)

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Reidemeister Move III can be

shown to preserve

tricolourability in the same

way.

However, there a quite a few

more cases to consider, so

lets skip this (and just believe

me that it works!!!!)

Knot Theory

What Is a Knot


TricolourabilitySupercoiling

Knot Colouring

Reidemeister Moves

ExerciseKnot Arithmetic

Conclusion


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So if one projection of a knot is tricolourable, then all

projections are.

So, for example, the trefoil can not be the same as the

unknot, because one is tricolourable, and the other is not.

However, the figure-eight knot is also not tricolourable - but

this does not mean that the figure-eight is the same as the

unknot.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Conclusion


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projections are.





unknot.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Conclusion


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projections are.





unknot.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Your Turn


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Now it is YOUR turn!

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Problem

determine which of the following knots are tricolourable.


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Solutions


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Time for the Solutions

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1

Th Fi t K t t b l


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The First Knot to be colour was:

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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Start by colouring any strand.

We have a choice of making

the two strands to the left and

right of the lower left crossing

either both red, or different

colours.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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Start by colouring any strand.

We have a choice of making

the two strands to the left and

right of the lower left crossing

either both red, or different

colours.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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Lets try making them both red.

Clearly we have a problem, as

we now must make the

remaining two strands red

also, and we have not used at

least 2 colours.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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Lets try making them both red.

Clearly we have a problem, as

we now must make the

remaining two strands red

also, and we have not used at

least 2 colours.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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So lets go back and make the

two strands different colours.

Now the crossing on the

furthest left has red and green

meeting at it, so the remaining

strand must be coloured blue.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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So lets go back and make the

two strands different colours.

Now the crossing on thefurthest left has red and green

meeting at it, so the remaining

strand must be coloured blue.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 1


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But we now have another

problem, as the remaining

strand cannot be legallycoloured.

There are no other ways to

colour this knot, and so it must

be NOT TRICOLOURABLE.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 2

The Second Knot was:


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 2


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Lets jump straight in and

colour three strands at a

crossing three different

colours.

Note that the crossing above

has red and green meeting,

and so must be coloured blue.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 2


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Lets jump straight in and

colour three strands at a

crossing three different

colours.

Note that the crossing above

has red and green meeting,

and so must be coloured blue.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 2


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Continue around the knot, and

we soon find that we have

coloured the whole knot.

So this knot is

TRICOLOURABLE

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 2


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Continue around the knot, and

we soon find that we have

coloured the whole knot.

So this knot is

TRICOLOURABLE

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3

The Third Knot was:


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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As before, lets pick a crossing

and colour the strands around

it with different colours.

The top-most crossing on theleft-hand-side has red and

blue meeting, and so the

remaining strand must be

coloured green.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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As before, lets pick a crossing

and colour the strands around

it with different colours.

The top-most crossing on theleft-hand-side has red and

blue meeting, and so the

remaining strand must be

coloured green.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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We now have a choice:

Colour one strand red, the

other blue.The same as above, but theother way around.Colour both strands green.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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In fact, all three options work.

So again, this knot isTRICOLOURABLE.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 3


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In fact, all three options work.

So again, this knot isTRICOLOURABLE.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 4

The Fourth Knot was:


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Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 4


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To save time, I will just tell you

the answers for the next three.This knot is NOT

TRICOLOURABLE.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 4


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To save time, I will just tell you

the answers for the next three.This knot is NOT

TRICOLOURABLE.

Knot Theory

What Is a Knot



Knot Colouring

Reidemeister Moves


Knot 5

The Fifth Knot was:


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Knot 5


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and this knot is NOTTRICOLOURABLE

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Knot 6

The Sixth and final Knot was:


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Knot 6


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As you can see, this knot isTRICOLOURABLE.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Return to knot 3


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Before I talk about the

telephone wire problem, lets

go back to the third knot onthe problem sheet.

Do you notice anything special

about this knot?

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Return to knot 3


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Before I talk about the

telephone wire problem, lets

go back to the third knot onthe problem sheet.

Do you notice anything special

about this knot?

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Two Trefoils


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It looks like two Trefoils stuck

together.

We say that this knot is a

composite knot, that it is thecomposition of two Trefoils,

and that the Trefoils are factor

knots.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Two Trefoils

I l k lik T f il k


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It looks like two Trefoils stuck

together.

We say that this knot is a

composite knot, that it is thecomposition of two Trefoils,

and that the Trefoils are factor

knots.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Composition of knots


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In general, we can compose

any knots.

To do this, we choose a strandon each knot that are on the

outside.

The we cut the strands and

splice the two knots together.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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any knots.


outside.



Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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any knots.


outside.



Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Composing with the Unknot


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Note that composition with the

unknot does not change the

original knot.Is the unknot composite?

Fortunately, the answer is no!

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Prime Knots


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We can also show that the Trefoil, Figure-Eight and most of

the other knots you have seen today cannot be written as

the composition of two knots.We call these knots prime knots.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Prime Knots


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We can also show that the Trefoil, Figure-Eight and most of

the other knots you have seen today cannot be written as

the composition of two knots.We call these knots prime knots.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

Analogue to the Integers

This all sounds familiar???

I l id i l h l i li i f i


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Its almost identical to the multiplication of integers:

We can multiply two integers together to get anotherinteger.

Multiplying by 1 does not change the number.Prime numbers are not the product of two integers (bothdifferent from 1).All integers can be written as the product of prime numbers(factors).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



It l t id ti l t th lti li ti f i t


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



It l t id ti l t th lti li ti f i t


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic



Its almost identical to the m ltiplication of integers


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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic





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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic





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Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The difference

One way the composition of

knots differs frommultiplication of integers is

that we have a choice of how


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to compose.

Note that the choice of strand

doesnt matter, by ambient

isotopy.

Our choice is which way round

we compose the knots -

compose with a knot that isupside-down could result in a

different knot.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The difference





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to compose.



isotopy.




different knot.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Knot Colouring

Reidemeister Moves

Exercise

Knot Arithmetic

The difference





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to compose.



isotopy.




different knot.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Other Invariants


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The story of the mathematics of knots continues in a similar

way, with more technical (but just as easy to calculate)

invariants, and more connections between knots and algebraicobjects (such as numbers).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Telephone wires

We shall now try to explain

why a telephone wirebecomes twisted in on itself.

We call this supercoiling.


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This problem also has (real)

practical use, as DNA also

supercoils.

The first thing to note is that a

telephone wire has a definite

thickness (unlike the knots we

have been looking at, wherethe thickness is not

considered).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Telephone wires





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e ca t s supe co g



supercoils.





considered).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Telephone wires





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p g



supercoils.





considered).

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Ribbons

To generalise a knot, weconsider knots in ribbons.

That is we shall assume our


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That is, we shall assume our

string has two edges and an

axis.

Note that the axis is a knot, as

we have studied so far.

So, a telephone wire forms a

knotted ribbon when the

receiver is down.

Knot Theory

What Is a Knot


Tricolourability

Supercoiling

Ribbons

Supercoiling

Ribbons

To generalise a knot, weconsider knots in ribbons.

That is we shall assume our

Knot Theory for Schools (Dec 07)

Documents

Transcript of Knot Theory for Schools (Dec 07)