Topics in Geometry: Introduction to Knot Theory

This is a math class about knots. While these are objects that arise relatively naturally in a variety of day-to-day and scientific settings, our challenge will be to carefully sort out how to describe, and ultimately study, knots using mathematical tools.

Here are some further remarks about this class:

• In a mathematical context, the study of knots falls into a broader area of research called topology, which is a branch of geometry (in a broad sense). However, topology (Math 426, for instance) is not a prerequisite for this course and as such we won't really take this point of view; our course will be more combinatorial in nature, building from the ground up.

• The course does, on the other hand, have prerequisites in the form of linear algebra (Math 221/223 or similar) and mathematical proof (Math 220 or similar). We will rely on both.

• This course will take the point of view that knots are a naturally occurring objects, and hence these objects are worthy of study in their own right. While I will draw on examples found "in nature" from time to time, this won't be the main emphasis of the course. In particular, if you are the sort of student who does not find it motivating to study and/or think about something for its own sake, this might not be the course for you!

To view the full content of this course, including lecture sumaries and assignments, turn off reader-view.

Mini projects

Project 1: Visualization aids
For this course, it will be helpful to have some sort of tool to help you visualize knots in 3-dimensions. This could be an old shoe lace, an extension cord, or (if you want to spend money) a tangle toy, for example. Try some different things. For this mini-project, all I want you to do is tell me what you plan to use.

Task: Prepare a short document that (1) introduces yourself, (2) says why you are interested in this class, and (3) includes a picture of the object you're going to use as visualization aid. Here's an example.

Due at: Submit your work by 5:00 pm on Friday January 11, 2019.

How to submit: Email a .pdf to our TA at mihmar(at)math(dot)ubc(dot)ca
Project 2: Knotting in everyday life
Knots show up all over the place. For this mini project, I want you to describe a knot or link that you find somewhere in science, nature, culture, or just your day-to-day life.

Task: Prepare a short document that (1) shows the knot or link you are going to discuss, (2) gives some background and context, and (3) identifies the knot in question using tools that we are developing (and references that we are using) in the course. Here's an example.

Due at: Submit your work by 5:00 pm on Friday February 1, 2019.

How to submit: Email a .pdf to our TA at mihmar(at)math(dot)ubc(dot)ca
Project 3: Knots and surfaces
Knots arise as the boundary of two-sided surfaces in 3-dimensional space. Conversely, given any knot Seifert gives an algorithm that can be used to construct such a surface. For this mini project, you'll construct such a surface using paper, scissors, and adhesives.

Task: Select a non-alternating 8-crossing knot from the table in the appendix in Adams (see also the Rolfsen knot table in Bar Natan's knot atlas). Implement Seifert's algorithm, and using paper and adhesives, construct the resulting Seifert surface. Be sure to colour your paper differently on each side, to ensure that the resulting surface is 2-sided.

Note: Group work is encouraged. You are welcome to work on your own, but you can work in groups of two people if you like – it will help to have an extra set of hands!

Due at: Submit your work by 5:00 pm on Monday March 11, 2019 (and then drop off your surface in class or at my office in the afternoon the next day).

How to submit: Email a photo of your surface to our TA at mihmar(at)math(dot)ubc(dot)ca

Important: Make sure to clearly indicate, in the body of your email, (1) who worked on the surface being submitted and (2) which knot you chose.
Project 4: Knots on campus

Task: We have seen that knots are closed curves in 3-dimensional space. For this project, you need to walk a circuit somewhere on campus (and carefully record what you did!) so that the closed curve you walked is non-trivially knotted. You will need to use buildings/stairwells/breezeways in order to introduce crossings. Notice that, having found a nontrivially knotted circuit on campus, a key challenge for this project will be finding a way to clearly describe the knot – your knot needs to be reproducible, and you need to identify the knot that you found.

Note: Group work is encouraged. You are welcome to work on your own, but you can work in groups of two people if you like.

Due at: Submit your work by 5:00 pm on Monday March 25, 2019.

How to submit: Email your our TA at mihmar(at)math(dot)ubc(dot)ca.

Important: Make sure to clearly indicate, in the body of your email, who worked on the project.

Lecture summaries

Note that you should ensure that you have lecture notes for all lectures. In the summaries below, Lectures marked with a ✫ indicate tose that contained material reaching beyond our primary resource (Adams).

✫Lecture 1: Some examples of knots
As a means of introduction, and for a first look at knot diagrams that will feature heavily in the course, we considered some examples of parametrized curves in 3-dimensional space having the form
x(t)=(2+cos(pt))cos(qt), y(t)=(2+cos(pt))sin(qt), z(t)=sin(pt)
for integers values of p and q.

For an instance of the objects from this lecture occurring elsewhere in geometry, you might read around a little about complex surfaces. For the same objects arising in nature, consider the rings of Saturn.

Reading: Adams, Section 1.1
Lecture 2: The Reidemeister Theorem

Now that we have a good working definition of a knot diagram (that is, a planar representation of an object that lives in three-dimensions), we need to be able to decide when two different knot diagrams represent the same knot. This is where the Reidemeister moves come in: we introduced three local moves on knot diagrams that don't change the knot represented. Reidemeister's theorem states that these three moves are enough, namely, that any two diagrams representing the same knot can be related to one another through some sequence of these three moves. It's important to note however that finding such a sequence can be difficult!

Reading: Adams, Sections 1.2 and 1.3
Lecture 3: Knot colouring

An outstanding issue, which we resolved today, is whether there exist any interesting (that is, non-trivial) knots. Our strategy was to define the notion of a tricolouring, and show that when a tricolouring exists for some diagram it has to exist on any diagram representing the same knot. This made an essential appeal to, and I hope illustrated the utility of, Reidemeister's theorem.

Reading: Adams, Sections 1.4 and 1.5
Lecture 4: Linking numbers

Today we introduced links, and in particular, defined the linking number between two components of a link. This is an integer-valued invariant, originally due to Gauss and having origins in electro-magnetism. We checked that it is an invariant; you should think about why the definition we gave will always be an integer.

Reading: Adams, Section 1.4
✫Lecture 5: Unknotting

Some knot invariants are easy to define (and really natural), but extremely difficult to compute. The minimal number of crossing changes you need to make to convert a knot to the unknot, a positive integer-valued invariant first considered by Tait, is an example of this.

Reading: Adams, Sections 3.1, 3.3, and 5.1
Lecture 6: Bridge position

We introduced the bridge number of a knot, and looked at how to see the equivalence between Adams' diagrammatic definition of this invariant in terms of maximal overpasses and a more geometric one in terms of local maxima in 3-space.

Reading: Adams, Section 3.2
Lecture 7: Continued fractions and rational knots

We took some time to unpack Adams' conventions for rational tangles. Ultimately, our interest is in the ability to use rational tangles to classify 2-bridge knots.

Reading: Adams, Sections 2.2, 3.1 (revisited), and 3.2
Lecture 8: The bracket polynomial

Thinking about tangles — atomic pieces of link diagrams — leads one to think about to other locally-defined operations on diagrams. This will be our next step: build polynomial invariants of knots and links. Our first pass at this was to derive the Kauffman bracket of a knot diagram.

Reading: Adams, Section 6.1
Lecture 9: The Jones polynomial

We have now defined a polynomial knot invariant that was first discovered in the 80s. This isn't the first polynomial but is an important one: its introduction opened up a lot of interesting lines of research. We proved that this invariant allows us to distinguish between the trefoil and its mirror image. Before the the Jones polynomial, proving this required quite a bit more machinery than we have access to in our course.

Reading: Adams, Section 6.1
✫Lecture 10: The skein module

We started by reviewing the calculation of the Jones polynomial of the trefoil, this time decomposing it into tangles. This led to our main divide-and-conquer technique for calculting the bracket polynomial (and, ultimately, the Jones polynomial): we saw that the bracket of a tangle could be viewed as an element in a vector space-like object called the Kauffman bracket skein module.

Reading: Adams, Section 6.1
Lecture 11: Alternating knots

The following fact can be proved using the Jones polynomial: any two reduced alternating diagrams representing the same knot have the same number of crossings.

Reading: Adams, Section 6.2
Lecture 12: The Alexander polynomial

A second, older, polynomial invariant of knots was introduced today: the Alexander polynomial is a different polynomial invariant of knots that has somewhat more geometric origins. In particular, as we'll explore next time, the degree of this polynomial tells us something about two-sided surfaces that have a given knot as their boundary.

Reading: Adams, Section 6.3
✫Lecture 13: Surfaces, linking matrices, and the Alexander polynomial

The Alexander polynomial shows up in other ways. Today we saw that, given a 2-sided surface in 3-space with boundary a knot, there is a linking matrix LK with the property that det(LK-tLK^T) returns the Alexander polynomial, up to a possible shift.

Reading: Adams, Sections 4.1, 4.2, and 4.3.
Lecture 14: Seifert surfaces

Given a knot diagram, Seifert gave an algorithm for producing a 2-sided surface with boundary that agrees with the knot. We carried this out for the figure 8 knot with paper, scissors, and tape.

Reading: Adams, Sections 4.1, 4.2, and 4.3.
Lecture 15: The HOMFLY polynomial

There is a 2-variable polynomial invariant of knots an links, which simultaneously generalizes the Jones polynomial and the Alexander polynomial. We calculated some examples and then verified the behaviour of this polynomial under taking mirrors.

Reading: Adams, Sections 6.3 and 6.4.
Lecture 16: More HOMFLY polynomial (Lecture by Claudius Zibrowius)

More on the HOMFLY polynomial, including connect sums and some parts of the proof of invariance.

Reading: Adams, Section 6.3.
✫Lecture 17: HOMFLY vs. Jones and Alexander

We saw that whitehead doubling gives knots with trivial Alexander polynomial. We constructed examples of pairs knots that have the same Jones polynomial but different HOMFLY polynomial.

Reading: Adams, Section 6.3.
Lecture 18: Knots as closed braids

A theorem of Alexander says that every link can be represented as the closure of a braid. We introduced braids today, and then went through an algorithm that establishes this result.

Reading: Adams, Section 5.4.
Lecture 19: The braid group

Braids are a little more organized than knots: Today we saw how to form the product of two braids, and how this turns the set of braids into a natural algebraic structure called a group.

Reading: Adams, Section 5.4.
✫Lecture 20: Artin combing

Pure braids are those with associated permutation that is trivial. Pure braids can be put in to a nice form – the Artin combed form – which gives rise to an algorithm for detecting the trivial braid.
✫Lecture 21: The Temperley-Lieb algebra

We revisited or calculation of the bracket polynomial, but applied to braids. This gives rise to something called the Temperley-Lieb algebra, and a somewhat more algebraic definition of the Jones polynomial.
✫Lecture 22: Markov's theorem and the Jones polynomial.

Markov's theorem tells us a series of moves between braids having equivalent closures. This was a key tool in the original definition of the Jones polynomial.

Topics in Geometry: Introduction to Knot Theory

Math 309

Winter 2019

Mini projects

Homework assignments

Lecture summaries