# Knots in three space: non-ambient isotopy equivalence classes

Here, our objects will be knots. Now I don’t want to be too pedantic but we do need some precision here. I will NOT worry about orientation as that will not be important for this type of knot theory.

We have two knots, $K_1 = g(K)$ where $K$ is, say, the standard circle in the plane and $g$ is a embedding (one to one continuous map), and $K_2 = h(K)$ is possibly some other knot.

We say that $K_1$ and $K_2$ are isotopic if there exists an isotopy $F: S^1 \times [0,1] \rightarrow S^3$ where $F(K,0) = g(K) = K_1$ and $F(K,1) = h(K) = K_2$. As usual, we require $F(_,t)$ to be an embedding for all $t \in [0,1]$ (if we allow for non embeddings, then we have a homotopy, not an isotopy).

NOTE it is very common in the literature to denote “ambient isotopy” by “isotopy”; usually it is clear from context what is meant. But beware of this practice.

Now in this setting (isotopy, possibly non-ambient), it IS important to note if the isotopy $F$ is smooth, piecewise linear, or merely topological.
1. If we insist that $F$ is smooth, then it is a theorem of differential topology that these smooth isotopies of compact sets “can be covered by an ambient isotopy”; that is, given a smooth $F$ we can find an ambient isotopy $G$ that accomplishes the same deformation. Note: this is only true for compact sets (in our case, knots). This is NOT true for lines in space.

2. If we insist that $F$ be piecewise linear (roughly speaking: take polygonal objects to polygonal objects), then the classification of knots is rather boring. Here is why: the following process is the result of a piecewise linear isotopy:

Think of shrinking the local knot with time until it disappears. Now remember that we are working in $S^3$; the complement of a piecewise linear ball is another piecewise linear ball on “the other side”.

Now one can find an interesting theory for links (knots with more than one component) using the piecewise linear isotopies; basically it is the usual ambient isotopy classes for links with the “local knots” removed.

3. If we allow for the isotopy to be topological, we CAN get interesting things happening, provided our knots are wild….”very wild”.
Now if the knots in question “pierce a disk” at a point (that is, there is a disk whose boundary links the knot and whose interior touches the knot “honestly”; that is, the knot enters the product neighborhood of the disk in one side and exits it out the other side; (to you experts: the Bing approximation theorem says we can use a disk that is p. l. everywhere but the point of entry).

Now if the knot pierces a disk: we can thicken that disk to “stretch out” the point into a nice, smooth arc, and then we can employ what we did earlier:

So if there is a knot that is NOT isotopic to the unknot (a simple, smooth simple closed curve that projects to a curve with no crossings), that knot has to be wild enough to fail to pierce a disk at any of its points.

Do such beasts exist? Yes; we’ve seen one in the previous post:

One obtains a knot by iterating this knotted solid tori construction over and over again, and taking the intersection. Yes, one does get a knot out this, and yes, this requires proof. R. H. Bing came up with this example: A Simple Closed Curve Which Pierces no Disk, J. Math. Pures and Appl. (9) 35, (1956), 339.

You can find that paper reproduced here in Bing’s “complete works.

Open Problem
It is unknown if this example is isotopic to the standard smooth unknot and it is unknown if there is ANY knot that is isotopic to the standard smooth unknot!

Is is known that there ARE simple knots which fail to pierce a disk at any of their points but ARE isotopic to the unknot:

see M. Brinn: Curves Isotopic to Plane Curves in Continua, Decompositions and Manifolds, Ed. Bing, Eaton, Starbird, University of Texas Press, 1980, pp. 163-166. Note: this article is mostly diagrams.

# What is Knot Theory Anyway?

A knot is an embedding of the circle $\{(x,y)|x^2 + y^2 =1 \}$ into 3-space. By “3-space” we usually mean $R^3$ or $S^3$, which is the 3-sphere, which can be thought of as $R^3$ with a point added at infinity. $S^3$ 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 $t \in [0, 2\pi),$ then $f(t) = (x(t), y(t), z(t)), x(t) = (2+cos(3t))cos(2t), y(t) = (2+cos(3t))sin(3t), z(t) = sin(3t)$ 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 $x, y$ 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 $S^3$ or $R^3$. There is knot theory of similar embeddings into other 3-manifolds, embeddings of $S^2$ into $S^4$ (higher dimensional knot theory) or embeddings of graphs into $S^3$.

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 $S^3$.

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 $[0,1]$ 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 $D^2 \times R$ where $D^2$ 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 $f:X \rightarrow Y$ is said to be proper if for all compact sets $C \subset Y, f^{-1}(C)$ is compact. Here is an example of a non-proper embedding: consider $f: R \rightarrow R$ given by $f(x) = arctan(x)$. The inverse image of $[0, \frac{\pi}{2}]$ 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, $K_1, K_2$; we say that $K_1$ is equivalent to $K_2$ if there is a map called an “ambient isotopy” that connects the two. More particularly there exists $F: S^3 \times [0,1] \rightarrow S^3$ where:
1. $F(-,t)$ is a homemomorphism of $S^3$ for all $t \in [0,1]$.
2. $F(K,0) = K_1$ and $F(K,1) = K_2$ for some $K \subset S^3$, $K$ homeomorphic to the circle.
The above is just a fancy way of saying that we can “deform space” to turn $K_1$ into $K_2$; almost never do we worry about finding, say, a formula for $F$.

It turns out that this definition is equivalent to the following simpler definition: $K_1, K_2$ are equivalent knots if there is some orientation preserving homeomorphism $f: S^3 \rightarrow S^3$ such that $f(K_1) = K_2$. 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 $K_1, K_2$ can be declared to be equivalent if there is a homeomorphism (possibly non-orientation preserving) $f: S^3 \rightarrow S^3$ such that $f(K_1) = K_2$.

If $f$ is orientation reversing and $f(K_1) = K_2$ then $K_1$ and $K_2$ 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: ($8_{17}$)

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.

Wild knots
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.

# First Post: Welcome!

This blog is an experiment that is designed to make me keep my own research on track.

What I hope to do: I hope to study the mathematics of

1. Embeddings of circles and arcs into 3-space; I am interested in those that are NOT the result of a smooth map.
2. Embeddings of the real line into open 3-manifolds: these beasts are lines embedded in such a way that the infinities of the real line goes to the ends of the manifolds.

I’ll say more in my next post; here are some objects of interest.