Knots, Braids, and Möbius strips - University of...

Knots, Braids, and Mobius strips

Thomas Prince

Magdalen College, Oxford

24 February 2018

Thomas Prince (Magdalen College, Oxford) Knots and Mobius strips 24 February 2018 1 / 19

Introducing Geometry

What does the word geometry mean to you?

Can you think of any results in Geometry you’ve covered in class?

Pythagorus’ theorem,

Circle theorems,

Trigonometry.

Circle theorems,

Trigonometry.

Circle theorems,

Trigonometry.

A little history

The geometry you will have seen in classes is some of the most ancientmaterial taught in schools (including in history classes).

There is a clay tablet from around 1800BC listing a collection oftriples a, b, c such that a2 = b2 + c2.

Euclid’s Elements written around 300BC contains many results oncircles, cones and cylinders that are well known to you.

Here are some faces associated to results you will know...

A little history

An Introduction to Topology

Today we look at an important area of modern geometry: topology.

Topology

Topology is the abstract study of shape, which is an active field ofresearch today. To a topologist shapes are ‘the same’ if they can bedeformed into each other by any ‘smooth’ process (no cutting or tearing),if you open a book these deformations are called ‘homotopies’.

Topology

Surfaces

One consequence of making any shape you can deform into another one‘the same’ is that you probably haven’t seen that many examples. Howmany topologically different shapes can you think of?

A list of shapes (up to deformation)

point (line?),

circle (square, triangle),

torus (bagel),

sphere,...

The last two can be made by gluing together a rectangle in certain ways.Can we see how? Can we see why we call these two-dimensional, orsurfaces?

Surfaces

point (line?),

torus (bagel),

sphere,...

Surfaces

point (line?),

torus (bagel),

sphere,...

Shapes in different dimensions

You’ll notice that trying to describe shapes in three dimensions gave usmuch more flexibility than we had in two dimensions. In fact geometrydepends a great deal on dimension one is working in. Some of the mostfamous/important results in geometry focus on this point.

Symmetry

One example of this is how the notion of rotational symmetry changesbetween two and three dimensions. What is a rotation in 2D? What is arotation in 3D?

Shapes in different dimensions

You’ll notice that trying to describe shapes in three dimensions gave usmuch more flexibility than we had in two dimensions. In fact geometrydepends a great deal on dimension one is working in. Some of the mostfamous/important results in geometry focus on this point.

Symmetry

One example of this is how the notion of rotational symmetry changesbetween two and three dimensions. What is a rotation in 2D? What is arotation in 3D?

Dirac’s belt trick

One nice illustration of the subtlety of working in three-dimensional spacerather than two was devised by Paul Dirac.

Belt trick

There are two experiments to do, both involve fixing one end of yourribbon securely:

Rotate the free end 360◦ (two half twists). Check that you cannotmake the ribbon untwisted without rotating the unsecured end again.

Rotate the free end 720◦ (four half twists). What happens now?

Dirac’s belt trick

One nice illustration of the subtlety of working in three-dimensional spacerather than two was devised by Paul Dirac.

Belt trick

Rotate the free end 360◦ (two half twists). Check that you cannotmake the ribbon untwisted without rotating the unsecured end again.

Rotate the free end 720◦ (four half twists). What happens now?

Mobius strips

Our first exercise will be all about a particular surface in three dimensionalspace.

Mobius strips

Take a strip of paper, rotate one end by 180◦ and glue the endstogether.

How many edges does this shape have? How many faces (sides)?

Mobius strips

Our first exercise will be all about a particular surface in three dimensionalspace.

Mobius strips

Take a strip of paper, rotate one end by 180◦ and glue the endstogether.

How many edges does this shape have? How many faces (sides)?

Cutting Mobius strips

Now you’ve seen a demonstration of what can happen when you cut aMobius strip we can work on our first project. You will need to make anumber of Mobius strips for this exercise.

Cut a Mobius strip in half lengthways (as shown) now cut theresulting object in half in the same way, what do you get?

Start cutting along the length of a Mobius strip 1/3 of the way upand continue cutting until you reach the point you started, what haveyou made?

Instead of making a Mobius strip using one half twist, what happensif you use two, three, four or five half twists.

Can you see any patterns in what you make, can you explain yourfindings logically?

Now you’ve seen a demonstration of what can happen when you cut aMobius strip we can work on our first project. You will need to make anumber of Mobius strips for this exercise.

Cut a Mobius strip in half lengthways (as shown) now cut theresulting object in half in the same way, what do you get?

Start cutting along the length of a Mobius strip 1/3 of the way upand continue cutting until you reach the point you started, what haveyou made?

Instead of making a Mobius strip using one half twist, what happensif you use two, three, four or five half twists.

Can you see any patterns in what you make, can you explain yourfindings logically?

Knots and Mobius strips

In fact we can see the connection between (certain) knots and Mobiusstrips without cutting it in half. Taking a Mobius strip made with an oddnumber of half twists and the boundary circle of that Mobius strip is aknot.

A knot in topology is a circle inside three-dimensional space. These havehad a key role in topology for the last forty years, they are also a perfectexample of a mathematical field: you start from something ‘easy’ anduncover a rich and unpredictable structure.

We say two knots are the same if you can move one onto the otherwithout breaking either. It isn’t at clear how many knots there could be.We’ll see in the rest of the session two more ways of making knots.

A Table of knots

New knots from old

All such tables of knots are incomplete, because there is always anoperation of making new knots from old ones.

Connect sum

Exercise: Make two Mobius strips made with three half twists and cutdown the middle (making a trefoil knot). We can attach them by cuttingthem open and forming one loop.

Knots which you can’t break into simpler knots are called prime.

New knots from old

Connect sum

New knots from old

Connect sum

Can we make all knots?

So we have two ways of making knots: taking the boundary of a Mobiusstrip and making connect sums. This makes infinitely many differentknots, but are there other knots (or not)?

Figure eight knot: 41

In fact there are (many) other knots, most of the ones in the table can’tbe made by taking the boundary of a Mobius band. The simplest suchknot is call the figure eight knot (or 41).

Making the 41 knot

In this exercise we will make a paper model of the 41 knot.

Making a figure eight knot

Take two strips and mark as shown below.

Put two half twists in strip 1 and glue the ends together.

Tape the A on strip 2 to the A on strip one.

Make two half twists in strip 2 and attach at the two Bs.

Crosscap number

Either thinking about the boundary of the shape you have made, or cuttingit apart (like you did for the Mobius strips) you have made a 41 knot.

Crosscap number

In fact we can make any knot using a procedure like this by attachingtwisted strips together and cutting. The minimum number of attachmentsthat need to be made has a name, the crosscap number, and is one way ofsaying how complicated a knot is.

Connection to Klein bottles

Given a surface (like your twisted Mobius strip, or glued strips) whoseboundary is a knot, you can make a closed surface (surface withoutboundary) by gluing a disc over the knot (note this will usually selfintersect and intersect the original surface, so it’s hard to draw).

Closing up surfaces

If the knot has cross-cap number 2 (so we can glue it from two strips) thesurface we obtain by attaching a disc is a Klein bottle.

Applications of Knot theory

So far we’ve made models of a number of interesting an important knots.In fact this is a field with practical importance in science as well as in puremathematics. Possibly the most striking application is to biology, butthere are applications in quantum field theory, statistical mechanics,...

The End

Knots, Braids, and Möbius strips - University of...

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