College Math Teaching

July 1, 2015

Embarrassing gaps in my mathematical knowledge

Filed under: mathematician, topology — Tags: , — collegemathteaching @ 1:56 pm

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 R^3 , add a point at infinity to get S^3 (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 S^3 . 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),

torusknot

satellite knot complements (the knot can live inside the solid torus that is the product neighborhood of a different, mathematically inequivalent knot,

satelliteknot

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 S^3. And, I really never understood what was going on at all.

horo_fig8

I suppose that one can view the boundary of these balls (called “horospheres”) as one would view, say, the level planes z = k in R^3 ; those planes become spheres when the point at infinity is added. This is a horoball packing of the complement of the figure 8 knot; missing is the horosphere at z = 1 which can be thought of as a plane.

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.

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April 30, 2013

My research

Filed under: advanced mathematics, research, topology — Tags: — oldgote @ 9:18 pm

If you are interested, I’ll be posting on topics that deal with my current area of research here.

I am reproducing this post below but will NOT publish further installments on this blog:

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:

trefoil

trefoilproject

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.

trefoilDiagram

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.

BorromeanRings.svg

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 .

twowildknots

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.

foxartinarc

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.

someproperknots

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.

knotteddna

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.

project4

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.

Blue_Figure-Eight_Knot

Non-invertible knots exist; here is an example: (8_{17} )

8_17_Knot.svg

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!

wildkarcs

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:

bingsling

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. (note: I’ll post this on my research blog, not here).

April 27, 2013

Unsolicited advise to young professors at heavy teaching load universities: Go to Research Conferences anyway!

This is coming to you from Ames, Iowa at the Spring American Mathematical Society Meeting. I am here to attend the sessions on the Topology of 3-dimensional manifolds.

Note: I try to go to conferences regularly; I have averaged about 1 conference a year. Sometimes, the conference is a MAA Mathfest conference. These ARE fun and refreshing. But sometimes (this year), I go to a research oriented conference.

I’ll speak for myself only.

Sometimes, these can be intimidating. Though many of the attendees are nice, cordial and polite, the fact is that many (ok, almost all of them) are either the best graduate students or among the finest researchers in the world. The big names who have proved the big theorems are here. They earn their living by doing cutting edge research and by guiding graduate students through their research; they are not spending hours and hours convincing students that \sqrt{x^2 + y^2} \ne x + y .

So, the talks can be tough. Sure, they do a good job, but remember that most of the audience is immersed in this stuff; they don’t have to review things like “normal surface theory” or “Haken manifold”.

Therefore, it is VERY easy to start lamenting (internally) “oh no, I am by far the dumbest one here”. That, in my case, IS true, but it is unimportant.
What I found is that, if I pay attention to what I can absorb, I can pick up a technique here and there, which I can then later use in my own research. In fact, just today, I picked up something that might help me with a problem that I am pondering.

Also, the atmosphere can be invigorating!

I happen to enjoy the conferences that are held on university campuses. There is nothing that gets my intellectual mood pumped up more than to hang around the campus of a division I research university. For me, there is nothing like it.

This conference
A few general remarks:
1. I didn’t realize how pretty Iowa State University is. I’d rank it along with the University of Tennessee as among the prettiest campuses that I’ve ever seen.

2. As far as the talks: one “big picture” technique that I’ve seen used again and again is the technique of: take an abstract set of objects (say, the Seifert Surfaces of a knot; say of minimal genus. Then to each, say, ambient isotopy class of Seifert Surface, assign a vertex of a graph or simplicial complex. Then group the vertices together either by a segment (in some settings) or a simplex (if, in one setting, the Seifert Surfaces admit disjoint representatives). Then one studies the complex or the graph.

In one of the talks (talking about essential closed surfaces in the complement of a knot), one assigned such things to the vertex of a graph (dendron actually) and set up an algorithm to search along such a graph; it turns out that is one starts near the top of this dendron, one gains the opportunity to prune lower branches of the group by doing the calculation near the top.

Sidenote
The weather couldn’t be better; I found time over lunch to do a 5.7 mile run near my hotel. The run was almost all on bike paths (albeit a “harder” surface than I’d like).

April 1, 2013

Fun for my Facebook Friends

Filed under: advanced mathematics, knot theory, topology — Tags: , — collegemathteaching @ 9:59 pm

Fun question one: can anyone see the relation between the following three figures? Note: I made a (sort of subtle) mistake in one of them….the one where the graph lines are showing)

tangle

fundamentallink

order3knot

Fun question two (a bit harder):

What is the relation between these figures?

type1calc

infinprod

And for the win: what is going on here? (this is ambiguous)

bingsling

Ok, I’ll help you with the last one: imagine this process (one solid torus (think: doughnut or bagel) inside a larger one, and repeat this process (think: those Russian dolls that are nested). If you then take the infinite intersection, you get a simple closed curve (not obvious) that is so badly embedded, it fails to pierce a disk at any of its points (and certainly fails to have a tangent vector anywhere).

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