Welcome to the exhibit of Geometry and Knots.
Knots are so much a part of our everyday world that we take them for granted, but mathematicians
have found that studying even the simplest knots can lead into almost unimaginable spaces.
In this movie, we'll take you on a short guided tour of the world of knots from a mathematician's
perspective.
Let's begin.
This figure eight knot can be untied since its ends are free.
From a mathematician's point of view, this knot is no knot at all.
But if we join the ends to form a loop, we can no longer undo the knot without cutting
it.
Now that's a knot.
These three knots look different, are they?
Two knots that look different may be just different arrangements of the same knot.
Sometimes a knot that looks knotted really isn't.
Here are three intertwined loops.
If you remove any one of them, the other two fall apart.
Mathematicians call the union of several loops a link.
By rearranging this link, we can see that the three loops are equivalent.
Here is another symmetric form of the same link.
In this form, it's called the Baromian rings.
When we look at a knot, we don't usually think about the space around it.
But to a mathematician, the space around a knot is just as real as the knot itself.
Sometimes it's easier to understand something by looking at what it's not.
The complement of a set is what's left when you take away the points of a set.
So what's not a knot is called its complement.
In 1988, Cameron Gordon and John Lukey proved that the complements of different knots can
never be the same space.
So studying the complements of knots helps us tell whether knots are the same or different.
What is left when you take away the points of a knot or link from three-dimensional space?
What's left when you can't see the knot, not just because there's no matter there, but
because space itself doesn't extend to where the knot used to be?
We'll look first at a simpler picture.
What is life like in a space with a single line missing?
Or in a plane with a single point removed?
We'll remove the point by pulling it upward, stretching the plane into a cone which grows
sharper and sharper.
Finally, the point disappears off to infinity and the cone becomes a cylinder.
The point at the cone's tip is special.
It's called a cone point.
As the cone steepens, the radius of a circle about the cone point increases, though its
circumference stays constant.
Meanwhile, an outsider looking from above sees the picture unchanged.
We introduce an observer who lives inside the cone's surface and a few objects for
him to observe.
For the insider, light rays travel in straight lines, but the outsider sees those lines as
curving around the cone point.
To understand the insider's view, cut a cone made of paper and unroll it onto a plane.
Since unrolling does not distort the paper, lines which are straight for the insider will
be straight lines on the plane.
This wedge is a building block for the cone and is called a fundamental domain.
The two edges resulting from the cut represent a single line back on the cone.
Anything crossing one edge reappears at the other.
We can mend the cut by stretching and gluing the edges.
We get the outsider's view where light rays look curved.
The position of the imaginary cut is arbitrary.
These insiders, touring the cone in their car, don't notice as we move the cut to follow
them.
They look right through the cut and see what seems to be another copy of the wedge.
For special values of the cone angle, a whole number of wedges fit together neatly, but
usually the remaining gap is filled with part of a wedge.
Here's another way a cone surface differs from a plane.
In a plane, only one straight line connects two points, but on a cone there may be several.
There may even be straight paths from a point to itself.
We make our two-dimensional objects three-dimensional by stacking them up.
The stack of cone points forms a line, called a cone axis.
The fundamental domain for this space is a solid wedge.
Our inside observer sees many copies of the car at once.
Which is the real one?
They are all equally real.
But the light rays from the car reach the observer from several different directions.
Remember our question, what is life like in a space with a single line missing, or in
a plane with a single point removed?
As the cone angle decreases, the space we are looking at changes.
Sometimes the wedge-shaped fundamental domains click together perfectly, like a jigsaw, and
we get a symmetric picture.
When n fundamental domains fit exactly around the cone axis, we say the axis has order and
symmetry.
As the order increases, the missing line recedes into the distance.
In the limit, the missing line disappears off to infinity.
The inside observer sees an endless straight row of cars, which are all a single car seen
by light traveling different paths.
Our building block, or fundamental domain, has become an infinite slab, and the cone
is now a cylinder.
We see how life would look in a space with a single line removed.
Let's now turn to the question, what is life like in a space from which the Boromian rings
have been removed?
We place six cone axes with order 2 symmetry on the faces of a cube.
This cube will be a fundamental domain for our experiment.
Let's try and see what this space is from the outside.
Remember that the walls of a fundamental domain determined by an axis of symmetry should be
thought of as being glued together.
We first glue the walls containing red axes, and then efface them as they are no longer
necessary.
Notice how the blue axes are joined together into an ellipse.
We now glue the walls containing green axes.
This joins the red axes into an ellipse.
We have folded along four of the six faces of the original cube to form an ellipsoid.
All that remains is to fold along the blue axis, which has also become an ellipse.
To do this, we make the front and back hemisphere of the ellipsoid bulge up.
The green axes, too, have joined to form a green ellipse.
That's what we have now, the Boromian rings.
Now we've seen that the outside view of our fundamental domain is the Boromian rings.
What is the insider's view like?
Remember around each cone axis, the insider sees two copies of every object.
We first activate the red axes.
The image from our fundamental domain gets replicated in the next door cube by the front
axis.
And both of these images get replicated by the back axis.
And so on, until we have copies of the cube extending all the way to infinity in both
directions.
Next, turn on the green axes which face forward.
This creates another infinite row of cubes.
Turning on the other green axes reproduces these two rows to give four rows, and so on,
until we have an infinite horizontal plane of cubes.
Finally, when the blue axes are turned on, they make the two-dimensional pattern be repeated
in layers to fill up the whole space.
This is what it looks like to live inside the space created by the order two axes on
the sides of the cube.
We can also work out what happens when the Boromian axes have higher order symmetry.
For instance, if we want them to be order four axes, we must build a fundamental domain
with 90 degree angles along these axes.
We must modify our cube so it has right angles along its six axes.
Impossible, you say?
You may not have noticed it, but we're escorting you into Lobachevskian, or hyperbolic, geometry,
where this and many other things are possible.
This dodecahedron, in true hyperbolic perspective, has 90 degree angles between every pair of
adjacent faces.
When we look directly down on the red axis, we see that the faces meeting there make a
right angle.
We can glue three more copies of the dodecahedron around this axis.
We can move our viewpoint inside this figure.
Now we have four-fold symmetry around one axis.
Before exploring further, we'll remove the walls and change the shape and color of the
beams.
We can continue to add copies of the dodecahedron around each colored axis.
But let's do this for some of the available green, blue, and red axes.
Eventually, the copies of the dodecahedron fill space without overlap.
Just as we tiled ordinary space with cubes, we've tiled hyperbolic space with regular
dodecahedra.
Let's fly around a little in hyperbolic space to get a better feel for it.
Notice how quickly apparent size changes as we move.
This is one of the biggest qualitative differences between our everyday space and hyperbolic space.
This is what it looks like to live inside the space created by order four axes along
the edges of the dodecahedron.
By adjusting the angles at the colored axes, we can derive similar pictures for order five
symmetry, order six symmetry, and so on for all higher orders.
Notice that as the order of the symmetry increases, the colored axes of the dodecahedron grow
very short and move far away.
Finally, in the limit, the red, green, and blue axes have receded to infinity.
The resulting figure is called a rhombic dodecahedron.
The six colored axes have been transformed into six vertices at infinity.
To better understand the geometry of this shape, we add transparent walls to one copy
of the figure and rotate hyperbolic space around its center.
As the vertices at infinity pass behind our eye, we see interesting patterns.
From inside the rhombic dodecahedron, the link has become infinitely far away so that
light can never reach it.
This is the picture of the complement of the Baromian rings.
We have succeeded in bending light so that it continues forever without hitting the link.
This geometry is called a hyperbolic structure for the link complement.
According to theorems of Mostow, Marden, and Prasad, there is at most one hyperbolic structure
for any link complement.
According to a theorem of Bill Thurston, all knots and links, with some simple exceptions,
have complements that admit hyperbolic structures.
Therefore, knots and links are completely determined by pictures such as this one.
This concludes our guided tour.
We've had a glimpse of how mathematicians understand knots and links and the spaces
around them.