These slides would be paired with “chalk” and discussion, but here are the slides:
Yes, mathematics is a huge, huge subject and no one knows everything. And, when I was a graduate student, I could only focus on 1-2 advanced courses at a time, and when I was working on my thesis, I almost had a “blinders on” approach to finishing that thing up. I think that I had to do that, given my intellectual limitations.
So, even in “my area”, my knowledge outside of a very narrow area was weak at best.
Add to this: 20+ years of teaching 3 courses per semester; I’ve even forgotten some of what I once knew well, though in return, I’ve picked up elementary knowledge in disciplines that I didn’t know before.
But, I have many gaps in my own “area”. One of these is in the area of hyperbolic geometry and the geometry of knot complements (think of this way: take a smooth simple closed curve in , add a point at infinity to get (a compact space), now take a solid torus product neighborhood of the knot (“thicken” the knot up into a sort of “rope”) then remove this “rope” from . What you have left over is a “knot complement” manifold.
Now these knot complements fall into one of 3 different types: they are torus knot complements (the knot can live on the “skin” of a torus),
satellite knot complements (the knot can live inside the solid torus that is the product neighborhood of a different, mathematically inequivalent knot,
or the knot complement is “hyperbolic”; it can be given a hyperbolic structure. At least for “most” knots of small “crossing number” (roughly: how many crossings the knot diagram has), are hyperbolic knots.
So it turns out that the complement of such knots can be filled with “horoballs”; roughly speaking, these are the interior of spheres which are “tangent to infinity”; infinity is the “missing stuff” that was removed when the knot was removed from . And, I really never understood what was going on at all.
I suppose that one can view the boundary of these balls (called “horospheres”) as one would view, say, the level planes in ; those planes become spheres when the point at infinity is added.
But the internet is a wonderful thing, and I found a lecture based on the work of Anastasiia Tsvietkova and Morwen Thistlethwaite (who generated the horoball packing photo above) and I’ll be trying to wrap my head around this.
Just a reminder: a knot in is wild if there is no homeomorphism where is a smooth (or p. l.) knot. So, how does one create these things?
This is certainly not an exhaustive list. But the reason I am writing up such a list is I am interested in the following question: for what knots is the following true: there is NO homeomorphism where but there IS a homeomorphism .
Note: the Gordon-Lucke Theorem says that this cannot happen for non-wild knots. That is, tame knots are determined by their complements.
This can happen for some wild knots though. In fact, it is possible for an everywhere wild knot to have a complement which is homeomorophic to a knot with just one wild point.
So, here we go:
1. Infinite product and their limits:
The limit part comes from the following type of construction:
Start with a finite product knot, put a “follow” torus around it, then put in more of the same knot, and a second follow torus which is a satellite of the first knot and repeat this process infinitely often, adding more satellite tori each time.
Then weather or not you get a standard infinite product knot or a knot that is wild at every point depends on how one slides the tori around. In each case, the knot comes from the infinite intersection of the tori.
If the tori are all slid to a single section of the first tori and arranged in order, one gets the classical infinite product. If the knots are “spread out” as one moves to the smaller tori, then one gets the “everywhere wild knot”. See Shilepsky’s paper for details on the second kind of knot and Bothe’s paper for the first kind. I relate the two kinds of knots (thanks for a good referee’s suggestion!) in my paper.
2. Infinite product of tangles.
These are knots like those in the upper right hand corner. Note that these types of knots CAN contain a wild arc in the following sense: given the knot built as the union of two arcs joined at their endpoints, at least one of the arcs is wild. In the infinite product case, the knot is the union of two tame arcs.
I don’t know what the implications of having homeomorphic complements is here.
3. Intersection of a nested set of tori.
Again, I don’t know if these are determined by their complements or not; my guess would be “no”. In my paper, I come up with some invariants for these types of knots.
4. Death spiral knots. (my term for these)
The suggestion for construction is given above. Basically, one starts with the first stage of a type of knot discussed in part 3. But then in the first construction cylinder, one deletes the left most returning arc cylinder (keeps the spanning one). Then cut the torus at the left disk, narrow the cylinder to the left of the left disk, shrink it and glue it to the disk corresponding to the start of the spanning cylinder. Do this in a way that the smaller tori align.
Now the resulting knot is the set of points that lie in an infinite number of cylinders.
I REALLY don’t know what is happening with the complements here.
Eventually I want to talk about proper knot theory: this is the theory of proper embeddings of the real line into open three manifolds.
Note: a proper embedding is one in which the infinities of the real line go to ends of the manifold; the inverse image of the embedded real line intersect a compact set in the manifold has to have a compact pre image. Example of a non-proper embedding of the real line:
Let ; if is any compact set containing in its interior then is unbounded and therefore not compact.
So it would be helpful to have some nice open manifolds to work with and one of the nicest examples of such gadgets are the Whitehead Manifolds.
This diagram represents three stages of the construction.
Here is the first stage: (imagine thickening the knot inside the torus)
If one keeps iterating these embeddings to get nested solid tori where is situated in as shown, and then one gets which is called the Whitehead continuum. Then is the required manifold.
One can also view this as an increasing union as well: in the “simple” diagram think of the inside torus as stage 1 and the outer torus as stage 2; that is, the inner torus is and the outer is . Now there is a homeomorophism where . Then define (composed times). Then the Whitehead manifold is .
It is well known that is open, contractable, not homeomorphic to and that is homeomorphic to ; there is also a new result by Gabai that says that is the union of two copies of . The eventual goal of this note is to show that is homeomorphic to . To do that, I’ll do a bit of warm up and then work through this paper by Glimm, which can be freely accessed here.
Warm up: why does a smooth knot unknot in 4-space?
When we work through the proof that is homeomorphic to we will be using the “4’th dimension” to lift the inner torus so as to unwrap it and then put it back in a trivial fashion (the inner torus would lie in a ball of the outer torus) and a product of these things is easy to visualize. So we need some 4-dimensional intuition to get the pictures clear in our mind.
(to you experts: yes, I know that can be thought of a cellular in and therefore is homeomorphic to , but I am trying to get an intuitive visualization here)
So, let’s see how a knot unknots in . Sure, there is an easy way: let where is identified with the hyperplane in . Now pick a point and let be the cone of from . is a piecewise linear disk but is NOT locally flat (and therefore isn’t a famous “slice disk”) because the link of the cone point is a copy of the knot , but this is enough to show that unknots. But that is not very satisfying; let’s construct a nice locally flat disk instead.
Start with a knot diagram; the knot consists of a finite number of arcs in the plane with a finite number of “overpasses” which leaves the plane but stays in the hyperplane.
Here is a view from the top with the z-axis coming at you and the w-axis just somewhere else. 🙂
Now any p. l. (or smooth) knot can be changed into the unknot by changing a finite number of crossings (the above example: we can get the unknot by changing any of the crossings). The minimum number that have to be changed is called the “unknotting number”; in general this is hard to calculate. But that doesn’t matter for this exercise. That that means for us is that there is a homotopy for which ,
is the unknot and if is the collection of arcs of on the plane, for all . That is, this homotopy keeps the plane arcs fixed and moves the necessary overpasses into underpasses so as to effect the necessary crossing changes. Note: of course, the knot has to pass through itself in ; we do get double points at a certain stage. In fact, we might say this homotopy affects the coordinate; think of a homotopy that drives the “unknotting overpasses” straight down and fixes the rest of the knot. So let’s describe the homotopy (in ) in two parts: and then . Now the ambient isotopy in can be described by for points on (the set of arcs in the plane) and
for and .
So the upshot: this isotopy swings the offending overpasses into the next dimension and then returns them underneath the plane; in the final slice, the knot has been taken to the unknot.
So, what does this have to do with the Whitehead manifold? Look at the simple picture of the first stage: one does exactly the same isotopy described in to change the single crossing into an under crossing; this puts in in a trivial manner; hence we have a homeomorphism from to where the latter is a trivial torus pair (one torus embedded in the larger torus in a geometrically trivial way (the inner torus lies in a ball contained in the larger torus).
So we obtain a homeomorphism between the pair which is the identity on the boundary and the latter is just a pair of solid tori where lies in a ball in . The reference linked to above (Glimm) shows that such homeomorphisms can be pieced together to yield a homeomorphism and the latter is known to be homeomorphic to .
Basically, the factor gives one enough “room” to unlink the Whitehead type link in the torus.
This best is simply connected; in fact it is contractable. But it has a very weird property: if one, say, constructs a smooth simple closed curve by, say, running along the center line of one of the defining tori, the complement of this curve (the knot group, if you well) is NOT finitely generated! So this beast is NOT .
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, where is, say, the standard circle in the plane and is a embedding (one to one continuous map), and is possibly some other knot.
We say that and are isotopic if there exists an isotopy where and . As usual, we require to be an embedding for all (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 is smooth, piecewise linear, or merely topological.
1. If we insist that 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 we can find an ambient isotopy 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 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 ; 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.”
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.
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.
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.