Trefoil knots in algebraic geometry

You've probably seen a trefoil knot before. It looks something like this:

In this week's post, we want to explain a quick but cute fact: the trefoil knot naturally shows up when you try to graph \(y^2 = x^3\) over the complex numbers!

This observation is actually the beginning of a great story about the possible shapes defined by polynomial equations. We also think this story lives up to the blog's name: just by graphing some simple equations, you can discover a hidden trefoil knot inside of complex space!

Graphs of polynomials are usually straight lines

When mathematicians study shapes, there are typically two levels of analysis: the local level, where we study a small portion of a shape; and the global level, where we try to understand the whole shape at once. It's usually a good idea to start with the local view before going to the global view.

For example, let's suppose we want to understand the shape that \(y^2 = x^3 + x\) will make when we graph it. It can be helpful to start by getting a good idea of the local picture first. And fortunately for us, calculus is very helpful here: the graph of \(y^2 = x^3 + x\) will always, when you zoom in far enough on any point, look approximately like a line.

...but these graphs aren't always straight lines!

This straight line behavior is in contrast to what happens at \(y^2 = x^3 + x^2.\) On that curve -- which has a very similar equation! -- calculus predicts that, at \((0, 0),\) things should behave strangely: locally it should look like two lines crossing. And indeed, if we graph it, this is exactly what happens, as shown in the widget below.

The way to understand this two lines crossing is that, when \((x, y) \approx (0, 0),\) the number \(x^3\) is much smaller than \(x^2\) (think of how when you start with a small number like \(1/10,\) raising it to powers makes it even smaller: \((1/10)^2 = 1/100\) and \((1/10)^3 = 1/1000\)). So, since \(x^3\) is so much smaller than \(x^2,\) we can approximate our equation \(y^2 = x^3 + x^2\) by \(y^2 = x^2,\) and the graph of that is two lines crossing: the solutions to \(y^2 = x^2\) are \(y = \pm x,\) so the graph is just the union of the lines \(y=x\) and \(y=-x.\) If you happen to know calculus, we'll remark that implicit differentiation can make this more precise.

So, when you graph a polynomial, locally it usually looks like a straight line -- except near a handful of critical points, like the one above, where the local behavior of the graph can be far more complicated!

A cuspidal cubic

Some critical points can look even stranger than two lines crossing. Consider the equation \(y^2 =x^3.\) It is graphed in the widget below, and has a critical point at the origin; zooming in on that critical point, something strange happens: the graph looks like some kind of 'doubled line.'

This doubled line behavior is called a cusp, and the equation \(y^2 = x^3\) is often referred to as the cuspidal cubic. It will be the object of interest for the rest of this point.

The cuspidal cubic, over the complex numbers

The great thing about polynomial equations is that they don't just make sense over the real numbers, but in fact they make sense over every number system. So, we can graph \(y^2 = x^3\) over the complex numbers \(\mathbb{C},\) and try to compare to our real picture.

What do we mean by graphing \(y^2 = x^3\) over \(\mathbb{C}\)? The answer is a little hard to geometrically understand, but is algebraically quite simple: instead of looking for pairs of real numbers solving this equation, we now look at the set of all pairs \((x, y)\) where \(x, y\) are both complex numbers obeying \(y^2 = x^3.\) For example, \((-1, i)\) is a point on the complex graph which we cannot see on the real graph.

Complex graphs are hard to draw, because \(\mathbb{C}\) is 2-dimensional (you need to specify two real numbers to determine a=one complex number: its real part and its imaginary part); so pairs of complex numbers are four dimensional. The equation \(y^2 = x^3\) ends up defining a 2-dimensional subset of a 4-dimensional space. As with the graph over the real numbers, most of the time this shape looks boring locally: for any point \((x, y)\neq (0, 0)\) on our graph, the surface just looks like a flat 2-dimensional plane; multivariable calculus can allow you to predict this.

But near \((0, 0),\) the graph looks totally different! What happens there? That is -- what's the analogue of the cusp we saw over \(\mathbb{R}\) when we graph in \(\mathbb{C}\)?

To get a handle on this graph, we're going to express it algebraically; this algebraic expression of the surface we can then draw inside of 3-dimensional space, to give you a picture of the shape.

We want to understand solutions to \(y^2 = x^3\) near \((0, 0).\) When you have a complex number, there are generally two ways to write it: you could write it in Cartesian form as something like \(2 + 3i\) or \(4 - 5i,\) or you could write it in polar form. Polar form is just the analogue of polar coordinates for complex numbers; let's quickly review it.

Digression: Polar form

When you want to specify the location of a point in 2-dimensional space, Cartesian coordinates tell you that you should say how far right to move from the origin, followed by how far up to move.

However, there's a different way to describe the location: instead of saying how far right and up to move, let's just describe the straight line path to the new point. This requires two pieces of information: a direction to face (encoded as an angle \(\theta\)), followed by a distance \(r\) to move in that direction.

Polar coordinates and Cartesian coordinates are related by a simple formula, easily derived from trigonometry: if your polar coordinates are \((r,\theta),\) then your Cartesian coordinates are \(x = r\cos\theta\) and \(y = r\sin\theta.\)

Complex numbers are just 2-dimensional points with some extra algebraic structure; so, these polar coordinates go through for them, and we can write every complex number as \[z = r(\cos\theta + i\sin\theta).\]

The extra algebraic structure on complex numbers makes polar form particularly natural, though: the angle addition identities of trigonometry imply \[(\cos\theta + i\sin\theta) \cdot (\cos\phi + i\sin\phi) = \cos(\theta+\phi) + i\sin(\theta+\phi).\] If you know trigonometry but have never seen this formula, we highly recommend you work through it! It implies that when you multiply complex numbers, their angles add.

This behavior -- where multiplication turns into addition -- is exactly what exponents do: multiplying \(b^x \cdot b^y\) becomes \(b^{x+y},\) with the multiplication turning into an addition. It is thus tempting to wonder if \(\cos\theta + i\sin\theta\) can be expressed in the form \(b^{\theta}\) for some base \(b\); remarkably this is true, though we won't elaborate on why in this article -- if you've never seen it before, imagine this is just some convenient notation to remind us of the rule for multiplying complex numbers. Thus people often write the polar form of a complex number as \[z = re^{i\theta},\] where \(e^i\) ends up being the base used to express these complex numbers (here \(e \approx 2.718\) is called Euler's number).

Visually, the intuition of polar coordinates is still here: \(e^{i\theta}\) is just the complex number at angle \(\theta\) on the unit circle, and \(r\) just scales us up or down.

Solving our equation

Let's rewrite \(y^2 = x^3\) in polar form. If we set \(x = re^{i\theta}\) and \(y = se^{i\phi},\) then \(y^2 = x^3\) becomes \[s^2e^{2i\phi} = r^3e^{3i\theta}.\]

For two numbers in polar form to be equal, they must have the same radius and same angular part; so we can separate this one equation into the two equations \[s^2 = r^3\] and \[e^{2i\phi} = e^{3i\theta}.\]

This first equation \(s^2 = r^3\) is just our real number \(y^2 = x^3\) from before, because now \(s, r\) are real; although \(r, s\) are both positive (because they are distances), so actually it's only the top half of our cusp.

This second equation \(e^{2i\phi} = e^{3i\theta}\) is more interesting, though. It is an equation relating two variables \(\phi\) and \(\theta,\) where \(\phi\) and \(\theta\) are not numbers but angles: the subtle distinction is that the numbers 0 and \(360\) are distinct, but as angles \(0^{\circ}\) and \(360^{\circ}\) are the same.

So, we're solving for angles \(\phi\) and \(\theta\) obeying \(2\phi = 3\theta.\) In the same way that we graphed \(s^2 = r^3\) on the \((r, s)\)-plane, we can graph the equation \(2\phi = 3\theta,\) but it requires a bit more care.

The torus

The natural geometric object associated to a number is a line: a number gives a point on the line. Similarly, the natural geometric object associated to an angle is the circle.

The natural geometric object associated to a pair of numbers is the plane, because pairs of numbers give points on the plane. But what is the geometric object associated to a pair of angles?

The answer is actually a torus! A torus can be generated by taking one circle, and looping it around in a circle. Thus a point on the torus is specified by two angles: the first angle tells you where you are on the initial circle, and the second angle tells you how far to rotate that initial circle.

Torus knots

What does \(2\phi = 3\theta\) look like, graphed on the torus? Your computer can make this graph this parametrically; the parametric curve \[t \mapsto (\theta = 2t, \phi = 3t)\] (where \(t\) is now an angle instead of a number!) obeys \(2\phi = 3\theta,\) and this parametric curve is easy (at least for a computer!) to graph. Graphing it leads to the following picture:

You might not be able to see it yet, but this picture is actually the trefoil knot from before!

In this animation, you can see us removing the torus from the picture and then reorienting the camera, allowing us to see that the trefoil knot from before is the same shape that we just drew on the torus!

The cone on the trefoil knot

Putting all this together, what is the final shape of the graph of \(y^2 = x^3\) near \((0, 0)\), over the complex numbers?

Remember that we rewrote this equation as \(s^2e^{2i\phi} = r^3e^{3i\theta}.\) The equation \(s^2 = r^3\) -- solved over positive reals -- defined a simple curve; the equation \(e^{2i\phi} = e^{3i\theta}\) defined a trefoil knot. Putting these two equations together, the final shape looks like a trefoil knot which gets smaller and smaller.

Why? Well, the final pair \((x, y)\) is obtained by taking our point \((3\theta, 2\phi)\) on the torus, and then scaling the \(y\)-coordinate by \(s^2\) and scaling the \(x\)-coordinate by \(r^3.\) As we approach the origin, these scaling facotrs \(s^2\) and \(r^3\) (which, remember, are equal to each other) get smaller and smaller; so, we have copies of the trefoil knot which scale down as they get closer to the origin. What does this shape look like?

Digression: Cones

Before answering the question of our graph, let's take a moment to reflect on cones. You've probably seen a cone before, but we're going to describe the cone in a slightly strange way: a cone is what you get by taking a circle, and then creating more and more copies of it, each of which is slightly scaled down. The below widget hopefully illustrates to you what we mean by this. It might help to rotate the camera a little, to get a sense of the 3-dimensional picture.

Back to our graph

Why did we have this digression on cones? Because the cone and our graph of \(y^2=x^3\) near the origin are built in the same way: in the cone we start with a circle and then scale it down to a point; in \(y^2=x^3\) we start with a trefoil knot and scale it down to a point. Thus, near the origin, \(y^2=x^3\) is a cone on a trefoil knot! The final image can be seen in the below widget; we highly recommend you rotate it around to get a good view of this cone!

Epilogue: advanced algebraic geometry

This story about the trefoil knot appearing in the graph of \(y^2 = x^3\) was very inspirational to the great mathematician John Milnor. Starting from this observation, he attempted to understand: what do graphs of general polynomial equations look like in the complex numbers, at least locally?

He found a beautiful general story: in the simplest situation of two variable equations, the graph would always resemble either the plane, or the cone on some knot (but perhaps a different knot than the trefoil).

From here, the story was developed by many mathematicians beyond Milnor. Mark Goresky and Bob MacPherson developed stratified Morse theory, which allowed them to push Milnor's results to more complicated equations than just two variable ones.

And in the two variable case, the legendary algebraic geometer Alexander Grothendieck managed to uncover a part of the story Milnor couldn't finish studying: which knots show up in the graphs of complex polynomials? Milnor observed that, in all of his examples, the knots which showed up had a peculiar property; this peculiar property was that a certain knot invariant called the Alexander polynomial always had solutions which were roots of unity, but the precise meaning of this is not important. This property was quite peculiar, which led Milnor to wonder: were the examples he had calculated explicitly just special in some way he didn't notice, or did all the knots which show up in graphs of complex polynomials have this property?

Grothendieck found an incredibly beautiful proof that in fact, all the knots which show up do have this property. This theorem of Grothendieck -- called quasi-unipotence of monodromy -- was a real tour de force in algebraic geometry. Grothendieck, quite remarkably, proved this by first solving these polynomial equations in modular arithmetic, and then using the properties of the solutions in modular arithmetic to remarkably deduce something about the graphs over the complex numbers! This kind of argument has since been used in many other contexts to prove great theorems which somehow seem impenetrable to geometry alone.