A knot is an embedding of the circle into 3-space. By “3-space” we usually mean or , which is the 3-sphere, which can be thought of as with a point added at infinity. is sometimes preferred because it is a compact space.
Note: sometimes we focus on the image of the embedding itself (i. e., the geometric object) and sometimes we focus on the map, which includes information about orientation.
Example: If one has then is a knot. Here are two different MATLAB plots of the image:
The second is a projection of the image of the trefoil onto the plane. If we endow such a projection with “crossing information”, we call the image a diagram for the knot.
Here, the broken line indicates that the strand passes under another strand.
It is custom to insist on “regular” projections, which means that:
1. All “singularities” (points on the diagram which correspond to more than one point of the knot) are double points (there are no points where 3 or more strands of the knot’s projection meet)
2. All crossings are “honest” crossings; that is there are no “tangents” (places where the projection “kisses” another strand).
Note: one can think of a diagram as a “shadow” of the knot on a plane, provided one adds crossing information at all double points.
Now not all knots possess a diagram, but it is a known fact that all smooth knots (knots that arise from differentiable embeddings) and all picewise linear knots (knots whose image consists of a finite number of straight line segments glued end to end) have a projection.
Most of knot theory research deals with smooth or piecewise linear embeddings of the circle into or . There is knot theory of similar embeddings into other 3-manifolds, embeddings of into (higher dimensional knot theory) or embeddings of graphs into .
Also, link theory deals with multiple knots together.
The above shows the Borromean Rings, which are three linked knots, no two of which are linked to each other. This is a famous 3-component link.
This blog will mostly focus on the following:
1. non-smooth (and non-piecewise linear) embeddings of the circle into .
These two diagrams are of non-smooth (and non-p. l.) knots; we call these wild knots. Notice how the stitches and arcs get smaller and converge to a point? That point is called a wild point. I will give a precise definition later; for right now we’ll tell you that it is impossible to assign a tangent vector to those points in some well defined way.
2. An arc is the image of into 3 space. The mathematics of smooth (or p. l.) arcs in 3-space is pretty boring. Every smooth or p. l. arc “can be straightened in space” into a straight, boring arc.
On the other hand, the mathematics of wild arcs (think: non-smooth/p. l. ) is every interesting.
The above arc has two wild points (the end points) and can NOT be straightened out in space into a straight arc. We’ll make this concept clear a bit later in another post).
3. Straight lines (a copy of the real line) into open 3-manifolds; we will insist that the “two infinities” of the line go to the “infinities” in the manifold.
In the above, the reader is invited to think of the “line” being embedded in the space where is the standard 2-disk. Think of an infinitely long solid tube or cylinder (like a long pipe).
I will call this Proper Knot Theory; the term “proper” is a technical term, which I will explain here: a continuous map is said to be proper if for all compact sets is compact. Here is an example of a non-proper embedding: consider given by . The inverse image of is not compact.
Equivalence Classes for Knots
In most of knot theory, what is studied is NOT the knots themselves but their “equivalence classes”. For example: the first example of the knot we have had a very specific function to define it. However, if we were to say, take a strand of the knot and move it a little, we’d get a different embedding, but mathematically we’d want to think of it as being “the same as” the original embedding. This makes the subject much more doable. Besides, knot theory is studied mostly because it impacts the study of the topology of 3-manifolds: such spaces are modified by doing operations (called “surgery”) which are often defined as being done along some embedded circle: a knot. In many cases, the objected obtained doesn’t differ “topologically” if the surgery knot is changed by some “motion of space”.
The same principle often applies if a scientist is, say, studying a knotted molecule or DNA strand.
So we need to state the equivalence classes.
Classical Knot Theory (the kind most often done)
Note: sometimes oriented knots are studied (the diagrams have arrows) and sometimes the unoriented knots are studied (no arrows). Sometimes this makes a difference as we shall see later.
The above is an example of an oriented knot diagram.
The most common equivalence class used:
Given two knots (or links) in three space, say, ; we say that is equivalent to $K_2$ if there is a map called an “ambient isotopy” that connects the two. More particularly there exists where:
1. is a homemomorphism of for all .
2. and for some , homeomorphic to the circle.
The above is just a fancy way of saying that we can “deform space” to turn into ; almost never do we worry about finding, say, a formula for .
It turns out that this definition is equivalent to the following simpler definition: are equivalent knots if there is some orientation preserving homeomorphism such that . Needless to say, this is easier to state, but one loses the sense of taking a knotted piece of string and playing with it (which is what you are doing in the first definition).
There is also another type of equivalence that is used: two knots can be declared to be equivalent if there is a homeomorphism (possibly non-orientation preserving) such that .
If is orientation reversing and then and are called mirror images.
So, classical knot theory (the kind most often studied) boils down to four different kinds:
1. oriented knots; mirror images considered equivalent.
2. oriented knots; mirror images NOT automatically considered equivalent.
3. non-oriented knots, mirror images considered equivalent.
4. non-oriented knots, mirror images not automatically considered equivalent.
A knot that is different from a knot with the same image but with a different orientation (arrow direction) is said to be non-invertible.
A knot that is different from its mirror image is said to be chiral.
The trefoil knot: is chiral but invertible (you can reverse the arrows by an orientation preserving homeomorphism)
The figure 8 knot: is NOT chiral and is invertible.
Non-invertible knots exist; here is an example: ()
The astute reader might wonder: “hey, you didn’t say anything about your isotopy or homeomorphism being smooth, piecewise linear or merely topological”. It turns out that in classical knot theory, this is a settled foundational question and therefore unimportant (here and here).
However this issue does appear in other kinds of knot theory, including those we will be discussing.
A knot (link or arc) is said to be tame if it is equivalent to a smooth (or p. l.) knot (equivalence class of choice). If it isn’t, it is called wild.
Note: it isn’t always immediately obvious if an arc is wild or tame; for example, the arc in the upper left hand corner is wild (wild point is the left end point) whereas the the lower right arc (which has separate trefoil knots converging to an endpoint) is actually tame!
We will discuss this later; note that the “infinite trefoil” arc is just on the edge of being wild; were we to add on, say, a straight segment at the left hand endpoint and extend it any finite distance at all, the arc would become wild. That appears to make no sense at all (at first glance) but in a later post I will provide a proof.
We will study wild knots of various kinds; note: it is possible for a knot to be wild at ALL of its points. We’ll get to this in a later post; if you can’t wait, here is an example: consider the following picture, which is supposed to represent a nested series of solid tori, (think: a bagel or doughnut) which are nested inside one another. If we intersect all of these knotted up tori, we end up with a very ill behaved wild knot in 3-space; this knot is wild at all of its points:
I am running out of steam; so in our next installment I’ll talk about different types of equivalence classes for knots in 3 space and for lines (proper knots) in open 3-manifolds.