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Knots and Braids: An introduction

Julia Collins, PhD student

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Knots

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What is a knot?

A knot is a circle embedded in 3-dimensional space.

=

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Why in 3-dimensional space?

In 2 dimensions all knots are trivial because there is no spaceto make complicated knots.

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Why in 3-dimensional space?

In 4 dimensions all knots are trivial because any knot can beunknotted.

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Higher dimensions

The important thing is to have a differenceof two dimensions.

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Higher dimensions

The important thing is to have a differenceof two dimensions.

Definition

Ann-dimensional knot is an embedding ofSn in Sn+2.

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Higher dimensions

The important thing is to have a differenceof two dimensions.

Definition

Ann-dimensional knot is an embedding ofSn in Sn+2.

For example, ifn= 2, we have a 2-dimensional sphere in

4-dimensional space.

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Questions of a knot theorist

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Questions of a knot theorist

How can we tell if two knots are the same?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

How are two different diagrams of the same knot related?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

How are two different diagrams of the same knot related?What is a prime knot?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

How are two different diagrams of the same knot related?What is a prime knot?

What kind of surfaces does a knot bound?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

How are two different diagrams of the same knot related?What is a prime knot?

What kind of surfaces does a knot bound?

What is the topology of the knot complement?

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Questions of a knot theorist

How can we tell if two knots are the same?

How many knots are there?

How are two different diagrams of the same knot related?What is a prime knot?

What kind of surfaces does a knot bound?

What is the topology of the knot complement?

Does the knot complement admit a hyperbolic structure?

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

Smallest genus of a bounding surface

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

Smallest genus of a bounding surface

Knot polynomials: Alexander, Jones, Kauffman, HOMFLY

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

Smallest genus of a bounding surface

Knot polynomials: Alexander, Jones, Kauffman, HOMFLY

Homology theories: Floer, Khovanov

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

Smallest genus of a bounding surface

Knot polynomials: Alexander, Jones, Kauffman, HOMFLY

Homology theories: Floer, Khovanov

Quantum invariants

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Telling knots apart

Knot theorists use invariantsto distinguish knots. For example...

Crossing/unknotting numbers

Smallest genus of a bounding surface

Knot polynomials: Alexander, Jones, Kauffman, HOMFLY

Homology theories: Floer, Khovanov

Quantum invariants

Hyperbolic invariants

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Knots in real life

Knots come up in a surprising variety of places!

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Knots in real life

Knots come up in a surprising variety of places!In your DNA

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Knots in real life

Knots come up in a surprising variety of places!In dynamical systems

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Knots in real life

Knots come up in a surprising variety of places!In chemistry

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Braids

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

Wh b d?

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What is a braid?

A braid consists ofn strings travelling from left to right, notlooping back on themselves.

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B id d k ?

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Braids and knots?

Answer: Attach the end of the strings to the beginning without

introducing more crossings.

B id d k t ?

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Braids and knots?

Answer: Attach the end of the strings to the beginning without

introducing more crossings.

B id d k t ?

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Braids and knots?

Answer: Attach the end of the strings to the beginning without

introducing more crossings.

B aids a d k ots?

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Braids and knots?

Answer: Attach the end of the strings to the beginning without

introducing more crossings.

Why the knot theorists are happy

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Why the knot theorists are happy

Theorem

Braids form a group under concatenation.

Why the knot theorists are happy

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Why the knot theorists are happy

Theorem

Braids form a group under concatenation.

Why everyone else is happy

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Why everyone else is happy

Why everyone else is happy

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Why everyone else is happy

Theorem

The braid group on n strands is the group of paths of n particles

moving in 2 dimensions.

(Jargon: the braid group is thefundamental groupof theconfiguration spaceof n particles in R2.)

Braids in real life

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Braids in real life

Braids come up in a surprising variety of places!

In air traffic control

Braids in real life

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Braids in real life

Braids come up in a surprising variety of places!

On the sun

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Thank you for your

attention!