Torus knot signatures, Part 3

Today we are going to learn some basic theory about slice knots. (Don’t worry – the relationship between this and the first two posts will become clear in due course!)

Every knot can be constructed as the edge of a 2-dimensional surface.  For example, the trefoil:

A trefoil bounding a surface (Picture credit: SeifertView)

(The wonderful program SeifertView is free to download and will produce a surface for any knot you give it.)

You will notice that this surface has a hole in the middle of it.  It turns out that only an unknotted piece of string can bound a surface with no holes in it (such a surface is called a disc). But what if we had another dimension of space in which to play?  Would that affect the result?

The 4th dimension is hard, if not impossible, to picture, so let’s start with a 3-dimensional analogy.  If we have a curve in 2 dimensions that intersects itself, it creates a kind of ‘hole’ in space.  But if we push the curve into the third dimension then the intersection point goes away and so does the hole:

Pushing a loop into 3 dimension

The question is: is it possible to remove holes in surfaces by pushing them from 3 dimensions into 4 dimensions?  In the case of surfaces bounded by knots, the answer is: sometimes.  These special knots are called slice knots.

It is, in general, very difficult to decide whether a knot is slice or not.  Mathematicians have worked out one way of telling if a knot is not slice: compute a number, called a signature, and if it is not zero then the knot is not slice.  If you know a little bit about matrices, read on and I will tell you how to compute the signature.

First, take your knot and find a surface that it bounds. (This is called a Seifert surface).  Then, draw a curve on the surface around every hole that you see, like this:

Curves around all the holes on a surface

We want to make a matrix that contains all the data of how these different curves link and knot around each other.  First, give each curve a number, then push each curve off the surface in turn and calculate how many times it links with all the other curves.  We enter the linking number of the pushed off curve i with another curve j into the (i,j)^th entry of our matrix.  For example:

Pushing each curve off the surface in turn

For this knot, our matrix is

Now, whenever you have a square matrix you can compute a set of numbers called eigenvalues.  These are numbers λ that satisfy the equation Mv=λv, where M is our matrix and v is some vector. (If you would like to learn more about eigenvalues, take a look at the Wikipedia article.)

The signature of a matrix is

(the number of positive eigenvalues) – (the number of negative eigenvalues)

and it is a deep theorem that slice knots will always have signature equal to zero.

Next time we will learn about the signature of torus knots, and we will see how they are tied in to the number theory we learnt about in the first two parts of this blog-series.  Stay tuned!

EUSci seminar

EUSci is the Edinburgh University Science magazine, and this evening they held the first lecture in a new seminar series. Before the talks there was pizza and drinks and plenty of opportunity to chat to other scientists: G and I got talking to a master’s student in ‘Ecological Economics’ (or ‘Economics for hippies’ as he put it) and a new PhD student in Chemistry (she gets to play with big machines that do lots of squeezing).

There were three talks over the course of the evening, all fairly biologically based.  I was very impressed by the quality of the speakers, who were only graduate students, and I’d like to tell you a bit about the topics they introduced us to.

First up was Katie Marwick talking about ‘Cognitive Enhancement’.  There are some drugs on the market that are designed to help people with mental disorders, for example, Ritalin for ADHD and Donepezil for Alzheimers, but the same drugs administered to ‘normal’ subjects resulted in an increase in some cognitive functions, such as alertness, memory and awakeness.  These drugs appear to have no short-term side effects but the long-term effects of regular use are unknown. Katie raised a lot of ethical questions associated with these kinds of drugs:

• Should they be legalised so that you can get them without a prescription?
• Is it fair for some people to take them, for example, before an exam?  Is this any more unfair than giving some children a private education or raising them with more books to read as they grow up ?
• Should some people be forced to take these drugs?  For example, doctors on night shifts to prevent them making as many mistakes?
• Would taking these drugs alter your personality?  If you considered yourself an absent-minded person then would removing that trait change who you see yourself as?

Sheep on St Kilda

Next was Adam Hayward talking about ageing in wild sheep populations.  A particular sheep population on St Kilda, to be precise.  He spends his time researching how much of an effect the environment has on the aging process of an animal.  To do this, he measures the amount of a certain parasite in the faeces of the sheep to see how good the sheep are at fighting off disease.  Firstly, it turns out that female sheep are much better at fighting the parasites than males, and they also have a life span that is about twice the male one (15 years compared with 7 or 8 years, although castrated males can live up to 17 years!). Adam also found that sheep who had lived a better life got better at fighting the parasites as they got older, whilst those who had had particularly hard lives (usually through weather conditions) tended to have more parasites in their bodies as they got older.  So, some lessons for humans: eat well and live happily (and get castrated!) if you want to have a better old age!

Finally we had Sarah Kabani who does her research on the parasite causing African Sleeping Sickness.  The parasite has a hard life, since it must survive both in the stomach of the tsetse fly (where it is attacked by gastric juices) and in the human bloodstream (where it is attacked by the immune system, but at least has plenty of sugar to feed on).  Sarah studies the transition in the genome of the parasite as it changes from ‘fly state’ to ‘human state’, using some fairly impressive-sounding technology.  Apparently they can print all 8,000 of the parasite genes onto a single microscope slide, and then they can make the genes glow at various intensities to show which ones are being used at which times.  The hope is that they can identify the genes which are most crucial to the parasite’s survival and then create drugs which can target these particular genes.

I very much enjoyed listening to all the talks, not just to learn more about the science but also to learn how scientists in other areas carried out their research.  There is quite a difference between a mathematician sitting with a pen and paper (and perhaps a computer) and biologists who are examining faeces, testing drugs on patients or scrutinising tiny glowing dots of genes.  I look forward to the next EUSci seminar!

Torus knot signatures, Part 2

Let’s quickly recap what happened last time.  We started with two coprime numbers, p and q, and for every number n which was neither a multiple of p nor q we wrote n as

n=ap + bq

where 0<a<q.  We defined the function j(n) as +1 if b was positive, and -1 if b was negative (and zero if n was not an allowed number).  The function s(n) was then defined as the running total of the j’s; for example, s(3) = j(1)+j(2)+j(3).

The question was then: how do we predict the function s?  The graphs we saw in the last post showed that s had all sorts of bumps and wiggles that varied a lot for different values of p and q.  How should we go about discovering the pattern?

Let’s try condensing all the data of s into just one number.  Then we can compare the value of this number over lots of values of p and q.  The number I’m going to choose is ‘the area under the graph of s’.  That’ll be the area inside the ‘V’ shape of the graph, since s is always negative.  It is easy to set up a computer program to calculate this value for lots of different p and q less than 100.  Here is the result:

Value of the area under the graph of s for different values of p and q. Red=small...Bluer=bigger.

Don’t worry about the dark spots in the picture: those are the points when p and q weren’t coprime, so there was no graph to find the area of.  The exciting thing is the very regular colouring of the graph!  The values of the area are changing very predictably with p and q, despite the graphs themselves being very unpredictable.

What is this new ‘area’ function?  What is it really measuring and how can we find a formula for it?  Tune in next time for a deeper explanation of what is going on here…

Torus knot signatures, Part 1

I want to tell you about the current mathematics I’m working on, because it’s exciting, surprising, beautiful, deep, and easy to explain!  Everything that a good piece of maths should be.  I’m going to have to explain it in a few parts though, leading you through the different steps I had to solve to get to the solution.

Here’s a problem to get you started.  I’m going to give you two numbers, p and q, which are coprime.  That means they have no common factors.  For example, I can’t give you the numbers 9 and 12, since both of them are divisible by 3.  But I can give you the numbers 9 and 10 since 9=3×3 and 10=2×5, and there is no overlap. Ok.  Now I give you another number n between 1 and pq-1, such that n is neither a multiple of p nor a multiple of q.  Your task: take n and keep minusing multiples of p from it until you get a number that is a multiple of q.  Then tell me whether the multiple of q is positive or negative.

Let’s do an example.  Suppose p=3 and q=5.  I need to give you a number n between 1 and 14, so let’s start with the simple n=1.  First we try 1-3=-2.  Not a multiple of 5.  So we try 1-(2×3) = -5.  Success!  And -5 is negative.  Now let n=8.  Try 8-3=5.  First time lucky! And this time we got a positive number.

For particular numbers p and q, we are going to do this for all possible n.  So if p=3 and q=5, we have to find the answer for n=1,2,4,7,8,11,13,14.  If the answer is positive, put j(n)=1, and if it’s negative we put j(n)=-1.  So, for these particular values of n, the corresponding values for j are -1,-1,-1,-1,1,1,1,1.  Now we keep a running total, and call that number s.  So the values for s are -1,-2,-3,-4,3,2,1,0.  Actually, for reasons that I will explain later, the interesting number to look at is 2 times s.

Let’s draw a graph of the function 2s and see what it looks like.  Does it always go “minus minus minus plus plus plus”?  Well, for p=2, the answer is ‘yes’, no matter what value you choose for q.  Here’s the graph of p=2,q=25, with the values of n running along the bottom and the value of 2s running up the side.

p=2, q=25

Notice the lovely ‘V’ shape that it makes.  The values of s decrease until a certain point, and then increase again.

Let’s see what happens when p=3.  Here’s the graph for p=3, q=10:

p=3, q=10

Oh no, what happened?  The graph is all wiggly at the bottom!  Minus, plus, minus, plus – what’s going on?  It’s obviously not a proplem with the number 3, since our previous example of p=3, q=5 worked out fine.  Maybe we just picked a funny combination of numbers.  We had better try another pair and see what happens there.  How about p=7, q=16?

p=7, q=16

Even more wiggles!  Yikes, it seems that this function is going to be pretty unpredictable.  The pattern seems to be very different for every p and q.  How on earth are we going to get a formula that tells us what is happening?

Tune in next time for more clues…

Mathematics fable

Hello everyone, Haggis reporting again after far too long away!  The summer ended up being more about doing maths rather than talking about it, and as we speak I am heavily embroiled in the mysteries of torus knots and signatures, not to mention quadratic forms and the pesky prime 2.  Perhaps I shall have a rant about the number 2 one day…

Helen Jackson and Adam Brewster

But today I wanted to tell you about another science communication endeavour that is starting in the maths department this month.  We’ve been approached by the animation company Binary Fable, which consists of Helen Jackson and Adam Brewster, who have a great idea for a new project.  They want to make 3 short animated films which involve the main character using mathematics to solve a problem.  Alongside that, there will be a website which will contain follow-up information about the maths in each film, together with a series of challenges for the public to solve.

Helen and Adam are looking for some ‘mathematicians in residence’ to supply ideas for the films, to blog on the website and to help construct the challenges to go with the films.  Construction work on the animation will probably not happen until April, ready to get the first film out when the new academic year starts in September.

Sadly I’m going to be too busy working on my Möbius strip project with Peter, along with helping Julia finish her PhD, to be a main point of contact for this project, but hopefully I can throw in some cool ideas of knot theory and topology to get them thinking!  Who knows, maybe there’ll even be a way to link this project to the Möbius one!

It’s always great to find new ways of bringing maths to the public, so I am very excited to see how this new method will proceed.  Stay tuned for more news!

Opportunities to inspire

It looks like the summer and autumn are going to be busy times in terms of science communication! In addition to developing the non-orientability exhibition I’ve agreed to help out with the following:

• Kickstart workshop, Wednesday 22nd July. Children aged about 16 from Edinburgh and the Lothians will descend upon the Edinburgh universities for a week in order to experience higher education. Each day they have the opportunity to do workshops in a couple of subjects to see what they are like, and hopefully find something they’d like to study at university one day. I will be giving a 20-minute talk on Knot Theory (either braids or DNA) to persuade them that maths is beautiful and creative, as well as useful in the real world.
• Maths masterclass, October/November. Every year Heriot-Watt and Edinburgh Uni run a series of about seven ‘maths masterclasses’ for bright 13-year olds. The sessions aim to give students a feeling of what research-level mathematics is all about through talks by mathematicians and then more in-depth activities. This year there is a vacancy for a speaker, and through a fortunate encounter with Robert Weston at a dinner party, my name came up! It’s going to be a lot of work preparing two hours’ worth of material, but I think the rewards will be worth it.

But for the moment, back to this PhD thingy. My flatmate Julia needs all the help she can get…

Surgery update

So, it’s been a little while since I last wrote and many of you are wondering what has happened about my surgery illustrations. I am wondering about this myself, not having heard from Erica (the journalist) in nearly two weeks now. The latest news was that the editor was considering hiring a professional animator (they’re willing to pay him, but not poor Haggis!) but they wanted my pictures to give him an idea of what was needed. I duly sent them off, all annotated and everything, but have no idea what the animator is going to do with them or when this article is going to appear.

Seeing as I’ve made these pictures I may as well give you (my loving readers) a brief description of what surgery is to show them off again.

So, you start off with your favourite manifold. A manifold is a mathematical space or shape which looks flat if you zoom in on any particular bit of it. For example, a circle looks curved from a distance but if you zoom in far enough then it’ll look like a straight line. Similarly, we live on a sphere but walking about in our everyday lives we think we’re on a flat surface. (Some people actually still think we live on a flat earth!) My favourite 2-dimensional manifold is a torus (or doughnut):

A 2-dimensional manifold called a torus

To start off the surgery procedure we have to cut out a “disc times a sphere”. In this case we will cut out a 1-dimensional disc (which is a line) times a 1-dimensional sphere (which is a circle).

Circle times a line is a cylinder

This gives us the following picture:

Torus minus a cylinder

The other half of the surgery procedure is to glue back in a different “disc times a sphere” along the edge where the other one was removed. In this case we will glue in a 2-dimensional disc (which is a filled-in circle) times a 0-dimensional sphere (which is two points – the boundary of the 1-dimensional disc).

Disc times two points is two discs

Gluing these discs in gives us:

Gluing in two discs

Finally, because I am a topologist I can pretend that this shape is made of plasticine and I can squeeze and push it about however I like (so long as I don’t make any holes in it). Hopefully you will agree with me that the simplest way to display this shape (after a bit of squeezing) is a sphere:

A sphere

So what surgery has done is to take a shape with a hole in it (the torus) and change it into a shape with no holes in it (the sphere). This is the general idea of surgery theory: to try and make shapes simpler. We can do it with high-dimensional shapes too (in fact, this is where it is most useful!) but this is very hard to visualise.

I hope you have enjoyed my few small pictures, and I will post again when I know more about the Kervaire article.  Comments, as always, are welcome.

Illustrating surgery

Recently, at the conference in honour of Sir Michael Atiyah’s 80th birthday in April, a solution to a long -standing mathematical problem was announced.  The Kervaire invariant problem was a conjecture that stated (roughly!) that in high dimensions there are spaces which cannot be turned into spheres by a process called surgery.  Just like in the medical world, mathematical surgery is about cutting bits out of shapes and then gluing different things back in.  To everyone’s surprise, the conjecture turned out to be (mostly) wrong: in all dimensions apart from 2, 6, 14, 30, 62 and maybe 126, all spaces can be turned into spheres by surgery.  Those spheres might be very bumpy and weird, but they are spheres nevertheless.

Now, this little explanation of mine is very brief and doesn’t contain near enough detail for anyone mildly interested.  Thankfully, the journalist Erica Klarreich (of Nature and New Scientist fame) is about to write an article that will put everyone’s curiosity to rest.  The article will appear on the web, rather than in a paper publication, and will be aimed at telling the story of the Kervaire invariant problem to a general audience.

“What has this got to do with little Haggis?”, I hear you ask.  Well, Erica needs pictures for her article.  She asked one of the mathematicians who worked on the problem, Doug Ravenel…who asked his friend Andrew Ranicki…who asked his graduate student Julia…who asked her sheepy housemate…me!  Seeing as I had learnt a little about the program POV-Ray in order to make my Möbius strip pictures last week, they thought that I would be up to the task of drawing a picture of surgery.

I’ve been playing around with POV-Ray a bit this weekend and it’s not taken me long to come up with some ideas.  To illustrate surgery, I need to take a shape (I chose a torus, or doughnut), cut a bit out of it and then glue a different thing back in.  Here’s a quick picture I made of a torus with a bit cut out of it:

Next I need to figure out how to glue circles onto the exposed edges of the torus, and I’m done!  Honestly, this really didn’t take me long and the final version will probably look quite different to this, but I’m very proud of myself so far.  POV-Ray is such a powerful program and my skills are barely scratching the surface.

I shall keep you posted on my future sketches and when/where the article will appear!

Möbius strip exercises

One of the great things about the Möbius strip is that you can do simple things to it and get very unexpected results.

Make yourself a few Möbius strips out of paper. I usually chop up a piece of A4 paper into long strips about 4cm wide, joining the ends of each strip with a half-twist. Now get yourself a pair of scissors and let the fun begin!

1) First cut along the centre of the Möbius strip. What do you think will happen? Will there be two strips after the cut, or one? Will they be orientable or not? 2) Now do the same thing on another Möbius strip, but make your cut 1/3 of the way along the width of the strip. Is the result the same as before? If not, why not?

3) Make a Möbius strip where the ends are joined together with three half twists instead of one. Repeat experiment (1), cutting down the middle of the strip. What do you get?

I will post up the results and explanations in a few days’ time. Until then, I look forward to your comments and bafflements!

Non-orientability 1: Möbius strips

So. Non-orientability. It’s a long word. What on earth does it mean?

Let’s start with a word that we do know. Orient means “to determine the position of, in relation to the points of the compass”. To be oriented means that you know, for example, which way is north. But that’s quite an Earth-specific definition. If you were floating in space, how would you define ‘north’? Perhaps you could use the concepts ‘left’ and ‘right’, ‘clockwise’ and ‘anticlockwise’ to give someone directions. These words would always work, right?

Well, be prepared to have your mind blown! Mathematicians have discovered universes in which the concepts of ‘left’ and ‘right’ make no sense at all.

The simplest example of such a ‘non-orientable’ space is called a Möbius strip, and is constructed as follows. Take a long thin strip of paper and connect up the opposite ends with a single half-twist. You should have something that looks like this:

Möbius strip

The first interesting thing to note about this object is that it has only one side. On your piece of paper, take a pen and trace a line along the length of the strip; you will soon see that you arrive back where you started but on the other side of the paper. So we have already lost the concept of a ‘front’ and ‘back’. What about left and right?

Meet Flib. He is a 2-dimensional creature whose universe is a Möbius strip. Due to an unfortunate paint accident, he has to live with the fact that his right hand is blue. One day, Flib is feeling adventurous and decides to explore the rest of his universe. Dum de diddly dee, along and along he wanders, until eventually he finds himself back where he started. But what is this? His friends all notice that now his left hand is painted blue instead of the right one, whilst poor Flib is wondering why his friends have all turned into the mirror images of the ones he left behind!

This little story shows us that the notions of left and right are simply not well-defined on a Möbius strip. And the Möbius strip is not the only example of a non-orientable space. As the weeks pass, I will show you other examples (sometimes in higher dimensions!) and will ask the all important question: is our own universe orientable?