knot theory for schools (dec 07)
TRANSCRIPT
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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
When Are Two Knots the Same
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
When Are Two Knots the Same
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 Moves
Exercise
Knot Arithmetic4 Supercoiling
Ribbons
SupercoilingKnot Theory
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What Is a Knot
When Are Two Knots the Same
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 Moves
Exercise
Knot Arithmetic4 Supercoiling
Ribbons
SupercoilingKnot Theory
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What Is a Knot
When Are Two Knots the Same
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 Moves
Exercise
Knot Arithmetic4 Supercoiling
Ribbons
SupercoilingKnot Theory
Wh I K
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a Knot
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a Knot
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a Knot
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a Knot
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a KnotD fi iti f K t
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What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
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
What Is a KnotD fi iti f K t
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When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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 Knot
SDefinition of a Knot
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When Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
e t o o a ot
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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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When Are Two Knots the Same
Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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 KnotWhen Are Two Knots the Same
Definition of a Knot
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When Are Two Knots the Same
Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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 KnotWhen Are Two Knots the Same
Definition of a Knot
E l f k d T l
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When Are Two Knots the Same
Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
Example
For example:
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Definition of a Knot
E l f k t d T l
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
The Unknot
The simplest knot has no
crossings.
We call it the Unknot.
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
The Unknot
The simplest knot has no
crossings.
We call it the Unknot.
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
T i l bilit
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
Examples of knots and Topology
A Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
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Tricolourability
Supercoiling
p p gy
A Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
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Tricolourability
SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
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Tricolourability
SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
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Tricolourability
SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A B i f Hi f K
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Tricolourability
SupercoilingA Brief History of Knots
The Trefoil
The next simplest knot, afterthe unknot, is the Trefoil Knot.
We usually draw it, like in the
diagram (right).
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A B i f Hi t f K t
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y
SupercoilingA Brief History of Knots
The Trefoil
The next simplest knot, afterthe unknot, is the Trefoil Knot.
We usually draw it, like in the
diagram (right).
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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y
SupercoilingA Brief History of Knots
A different Trefoil
But what about this knot?
It is clearly the trefoil, and we
could easily remove the extra
twist.
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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SupercoilingA Brief History of Knots
A different Trefoil
But what about this knot?
It is clearly the trefoil, and we
could easily remove the extra
twist.
Knot Theory
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
S ili
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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SupercoilingA Brief History of Knots
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
S ili
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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Supercoiling
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a Knot
Examples of knots and Topology
A Brief History of Knots
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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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
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Supe co g
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
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p g
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
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48/199
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
8/2/2019 Knot Theory for Schools (Dec 07)
49/199
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
8/2/2019 Knot Theory for Schools (Dec 07)
50/199
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
8/2/2019 Knot Theory for Schools (Dec 07)
51/199
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
8/2/2019 Knot Theory for Schools (Dec 07)
52/199
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
What Is a KnotWhen Are Two Knots the Same
Tricolourability
Supercoiling
Definition of a KnotExamples of knots and Topology
A Brief History of Knots
-
8/2/2019 Knot Theory for Schools (Dec 07)
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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
What Is a Knot
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
H d f k
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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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move 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
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move II
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Reidemeister Move II
This move passes one strand
under or over another.
Or pulls one strand out from
under, or over, another.
It either creates or removes
two crossings.
Knot Theory
What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move II
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Reidemeister Move II
This move passes one strand
under or over another.
Or pulls one strand out from
under, or over, another.
It either creates or removes
two crossings.
Knot Theory
What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move II
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Reidemeister Move II
This move passes one strand
under or over another.
Or pulls one strand out from
under, or over, another.
It either creates or removes
two crossings.
Knot Theory
What Is a Knot
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move III
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Reidemeister Move III
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move III
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Reidemeister Move III
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Reidemeister Moves
Example
Reidemeister Move III
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Reidemeister Move III
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
When Are Two Knots the Same
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
Reidemeister MovesExample
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
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Example Time
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Example Time
The Knots look a little like this:
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step One/Six
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p
Use ambient isotopy.
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step Two/Six
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p
Use RMII (and ambient isotopy)
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step Three/Six
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p
Use RMII twice.
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step Four/Six
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p
Use RMIII.
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step Five/Six
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p
Use RMII (in reverse).
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
Step Six/Six
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Use RMI.
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Reidemeister MovesExample
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.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
Knot Arithmetic
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We shall now consider one of the simplest invariants -
tricolourability.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
Knot Arithmetic
Examples
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Lets try colouring some knots.
First, the Unknot.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
Knot Arithmetic
The Unknot
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The Unknot is not tricolourable,because we can only use one
colour.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
Knot Arithmetic
The Trefoil
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Next, The Trefoil.
Knot Theory What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
Knot Arithmetic
The Figure-Eight
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Finally, lets try and colour the
Figure-Eight Knot.
Knot Theory
What Is a Knot
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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
When Are Two Knots the SameTricolourability
Supercoiling
Knot Colouring
Reidemeister MovesExercise
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Reidemeister Move II
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Reidemeister Move II
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Reidemeister Move II
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Reidemeister Move III
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Reidemeister Move III
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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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Your Turn
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Now it is YOUR turn!
Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Problem
determine which of the following knots are tricolourable.
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Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Solutions
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Time for the Solutions
Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 2
The Second Knot was:
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Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 3
The Third Knot was:
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Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 3
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In fact, all three options work.
So again, this knot isTRICOLOURABLE.
Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 3
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In fact, all three options work.
So again, this knot isTRICOLOURABLE.
Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 4
The Fourth Knot was:
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Knot Theory
What Is a Knot
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
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
When Are Two Knots the Same
TricolourabilitySupercoiling
Knot Colouring
Reidemeister Moves
ExerciseKnot Arithmetic
Knot 5
The Fifth Knot was:
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Knot Theory
What Is a Knot
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Analogue to the Integers
This all sounds familiar???
It l t id ti l t th lti li ti f i t
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Analogue to the Integers
This all sounds familiar???
It l t id ti l t th lti li ti f i t
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Analogue to the Integers
This all sounds familiar???
Its almost identical to the m ltiplication of integers
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Analogue to the Integers
This all sounds familiar???
Its almost identical to the multiplication of integers:
-
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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
When Are Two Knots the Same
Tricolourability
Supercoiling
Knot Colouring
Reidemeister Moves
Exercise
Knot Arithmetic
Analogue to the Integers
This all sounds familiar???
Its almost identical to the multiplication of integers:
-
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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
When Are Two Knots the Same
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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that we have a choice of how
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
When Are Two Knots the Same
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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that we have a choice of how
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
When Are Two Knots the Same
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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that we have a choice of how
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
When Are Two Knots the Same
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
When Are Two Knots the Same
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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We call this supercoiling.
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
When Are Two Knots the Same
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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e ca t s supe co g
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
When Are Two Knots the Same
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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p g
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
When Are Two Knots the Same
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
When Are Two Knots the Same
Tricolourability
Supercoiling
Ribbons
Supercoiling
Ribbons
To generalise a knot, weconsider knots in ribbons.
That is we shall assume our