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 naturally occurring objects–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. Instead, this course is meant to provide a window to pure mathematics research, and hopefully provide a partial answer to the question "what do mathematicians do?"

A previous (online) version of this course can be found here.

Assignment submission and grading will make use of Canvas. Here are some essential guidelines for assignment submission:

1. Your solutions to assignments must be submitted using canvas. In particular, do not email assignments.

2. Your solutions must be submitted as a single file.

3. The only accepted file type is PDF.

4. Files should be named according to the following convention:
YourName_YourStudentNumber_AssignmentNumber.PDF

There are free online tools available that will allow you to complete (2) and (3). Due to the volume of data being managed, it is really important that these 4 points be met so that we can grade your work effectively. As a result, if one (or more) of these four points are not met, your assignment will not be accepted/graded.

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

Course Units

This course is divided into 7 units. Each unit has one homework assignment, to be completed and submitted online, that taken together will make up your final grade in the course.

Unit 1: Foundations
The relevant sections from Adams are 1.1 through 1.5.
Lecture 1 (January 9)
Prelude: parametrizing curves in the torus
As a point of entry for studying knots, we reviewed some material from calculus about parametrized curves. Specifically, we considered the torus as a surface of revolution (as seen in your calculus class) and, by restricting certain values of θ and φ, generated parametrized circles that live on the torus.

Lecture 2 (January 11)
Knot diagrams
While we have a reasonable definition of what a knot is, we don't yet have a good way of working with these objects yet. To this end, today we looked at some slightly more interesting curves on the surface, in order to get at what might be a good diagrammatic formalism for studying knots.

Lecture 3 (January 13)
An example or interest
After reviewing some of the material from last time, we looked carefully at a diagram for the trefoil knot. This led us to a reasonable definition of a knot diagram, which is a choice of planar representative for a knot. We now need to carfuly consider what sort of equivalence exists between diagrams that represent the same knot.

Lecture 4 (January 16)
Reidemeister's theorem
Deciding if two knot diagrams represent the same knot can be hard. Fortunately, there is a theorem, due to Reidemeister, that tells us precisely when this is the case: This gives a (short!) list of possible moves, and guarantees that two diagrams that represent the same knot will always be related by some collection of these moves applied in sequence. The list of required moves may be very long, and it is definitely not unique, so this theorem is hard to apply. But it is useful! We'll see how to use it to construct tools used to distinguish knots.

Lecture 5 (January 18)
Colouring knots
We've now shown that interesting knotting exits. We did this by finding a property of a knot diagram (tricolourability), unchanged under Reidemeister moves, that was enjoyed by the trefoil knot but not by the trivial knot.
Assignment 1 (linked here) is due Monday January 23 by 11:59pm.

Unit 2: Linking and knotting
The relevant sections from Adams are 1.4, 3.1, 3.3, and 5.1.
Lecture 6 (January 20)
Invariants of knots and links
We revisited the work from the last lecture in a more general context: a knot invariant is a rule F that assigns a value F(K) (a number, a polynomial, a ...) to a given knot K. This should satisfy the property that F(K)=F(K') whenever K and K' are equivalent knots. Invariants also exist to study linking, and we saw that the linking number between components of a link gave an integer-valued invariant.

Lecture 7 (January 23)
Crossing numbers
An example of an invariant that is easy to define but difficult to compute is the minimal number of crossings possible in a diagram representing a given knot. We saw that this value has the desirable property that it vanishes if and only if the knot is trivial, that is, the crossing number detects knottedness. Surprisingly, we showed that it is possible to say a little more: based on what we have seen to this point in the course we are able to prove that the trefoil knot has crossing number 3.

Lecture 8 (January 25)
Unknotting numbers
Another natural but very difficult to compute numerical invariants of knots is the unknotting number. In general, the unknotting number of a knot need not be realize on a minimal diagram, that is, a diagram realizing the crossing number of the knot. By contrast, amazingly, McCoy proved that when K is an alternating knot with unknotting number 1, any alternating diagram for K contains an unknotting crossing.

Lecture 9 (January 27)
Torus knots I: Guest lecture by Mihai Marian
A large supply of non-alternating knots are provided by torus knots, which we encountered in earlier lectures.

Lecture 10 (January 30)
Torus knots II: Guest lecture by Mihai Marian
A survey of known results pertaining to crossing number and unknotting for this class of knots. These results appeal to surprisingly deep mathematics!
Assignment 2 (linked here) is due Friday February 3 by 11:59pm. Deadline extended to Sunday February 5 at 11:59pm.

Unit 3: Knots and continued fractions
The relevant sections from Adams are 2.2, 3.1, and 3.2.
Lecture 11 (February 1)
Tangles
Tangles provide the building blocks for knots that Conway used for tabulation. Today we set this up.

Lecture 12 (February 3)
Rational and algebraic knots
Building on the tangles we constructed last time, we defined two classes of knots, following Conway: Rational knots (closures of rational tangles) and algebraic knots (closures of algebraic tangles). Conway's ideas for knot tabulation lead to some closely related (but distinct!) pairs knots related by an operation called mutation. Here is a link to a nice article in Quanta about Lisa Piccirillo's recent work on the subject. We'll come back to mutation in later units.

Lecture 13 (February 6)
Bridges
We introduced a third geometric/numerical invariant called the bridge number. This has a geometric definition in terms of local maxima along a knot, but it is computable by considering an equivalent definition that counts the number of maximal overpasses in a diagram.

Lecture 14 (February 8)
Rational knots, revisited
We saw that rational knots are equivalent to two-bridge knots, and more, we saw that rational tangles were in one-to-one correspondence with rational numbers, by using continued fractions to convert a rational tangle into a fraction.

Lecture 15 (February 10)
Housekeeping
Before moving to the next unit, I made some comments about bridges and the unknot, and spent a little more time on the example from last day comparing the equivalence of tangles and the equivalence of associated continued fraction expansions.
Assignment 3 (linked here) is due Friday February 17 by 11:59pm.

Unit 4: Polynomial invariants
The relevant section from Adams is 6.1.
Lecture 16 (February 13)
From diagrams to polynomials
Today we derived a divide-and-conquer strategy for converting a diagram into a Laurent polynomial with integer coefficients.

Lecture 17 (February 15)
Towards invariance
Building on our work from last time, we say that the bracket polynomial is almost an invariant of knots: the only point of failure is the first Reidemeister move. This led to a discussion on how we might recover from this disappointment; there are two useful solutions, as it happens.

Lecture 18 (February 17)
Brackets and the Jones polynomial
We found that the bracket polynomial is almost a knot invariant, and then defined a slight adjustment to get the Jones polynomial, which is a knot invariant.

Lecture 19 (February 27)
Trefoils
Returning from break, we reviewed some material by focussing on examples. By calculating the relevant Jones polynomials, we showed that the left- and right-hand trefoils are distinct knots. We also discussed how to use the bracket polynomial to 'see' this fact.
Assignment 4 (linked here) is due Friday March 3 by 11:59pm.

Unit 5: Applications and further structure
The relevant sections from Adams are 6.2 as well as parts of 6.3 and 2.3 (revisited).
Lecture 20 (March 1)
Orientations
We looked carefully at how the Jones polynomial depends on the orientation of a link. One thing we concluded from this discussion is that the Jones polynomial is insensitive to the orientation on a knot, even though an orientation is required in order to calculate it!

Lecture 21 (March 3)
Span: Guest lecture by Mihai Marian
We introduced the span of a polynomial and observed that, based on our work from last time, the span of the bracket polynomial gives us a new numerical invariant of knots and links. This number is useful! It can be used to prove that any two reduced alternating diagrams for the same knot have the same number of crossings.

Lecture 22 (March 6)
Alternating diagrams
Here's a key lemma: that the span of the bracket of a reduced alternating diagram is 4 times the crossing number of the diagram. We proved this today.

Lecture 23 (March 8)
Alternating diagrams, continued
We picked up where I left things at last time, and looked more closely at the final steps in the proof of the key lemma. Two of the important steps show up on the assignment for this unit.
I ended class with an aside on Trotter's proof that non-invertible knots exist (a question from last week), which is constructive and appeals to material from Math 426 (to build an invariant group) and Math 322 (to derive a contradiction to the existence of an inversion in the example Trotter produces). While this material reaches beyond the scope of our class, if you are taken/have taken either of these courses, you are equipped to read Trotter's original 1964 paper, which is available for free, and linked here .

Lecture 24 (March 10)
Mutation
Mutation is the result of removing a tangle, rotating it by 180 degrees, and replacing it back in the knot diagram. This process tends to change the knot type, but this change is very difficult to detect in practice. In particular, the bracket polynomial is unable to see the difference, and today we looked closely at why this is the case. Along the way we saw that, for example, moving a small ring from one component to another (within a given link diagram) is a process that goes undetected through the lens of the bracket polynomial. (Indeed, you can check that this ss a special case of mutation.) In order to formalize all of this we are led to consider the Kaufman bracket skein module for tangles, which we introduced today.
For people who would like to read more, a great reference is the paper The quest for a knot with trivial Jones polynomial by UBC's own Dale Rolfsen. You can find it through the UBC library, which has online access to the book Topics in knot theory containing the paper (it's on pages 195–210). It's an interesting read about a problem that is still open, and the discussion on skein modules is in the first few pages.
Assignment 5 (linked here) is due Friday March 17 by 11:59pm.

Unit 6: Surfaces and the Alexander polynomial
The relevant sections from Adams are 4.1 through 4.3 as well as parts of 6.3.
Lecture 25 (March 13)
The Alexander polynomial
We introduced a second polynomial invariants of knots: the Alexander polynomial. Perhaps surprisingly, it predates the discovery of the Jones polynomial by over 50 years.

Lecture 26 (March 15)
Primer on surfaces
A little bit of background on surfaces, before we dive into the interplay between knots and the surfaces that they bound.

Lecture 27 (March 17)
Seifert's algoritm
Every knot bounds an orientable surface in 3-dimensional space, and Seifert gave an algorithm for constructing such a surface that starts from a knot diagram.

Lecture 28 (March 20)
A triptych of examples: the trefoil
We worked through Seifert's algorithm on various projections of the trefoil, and compared this to our cut-and-paste construction associated with the trefoil from last time. This assignment of surfaces to diagrams leads to a definition of the Seifert genus of a knot: the smallest genus among all surfaces constructed for various choices of diagram for a given knot.

Lecture 29 (March 22)
Linking and the Alexander polynomial, revisited
Today we put our work on surfaces in conversation with the Alexander polynomial. We found a matrix, which tracks linking data from the surface, and used this to derive the Alexander polynomial by calculating a determinant. This is a little surprising, given how we came to it, but it hints at the geometric/topological origins of this polynomial going back to Alexander's original work. For those interested in learning more, I strongly encourage you to take Math 426 in the future!

Lecture 30 (March 24)
Example: 7₆
Seifert's algorithm, applied to the knot 7₆, yields a surface of genus 2. This means there are 4 bands attached to a disk in such a way that the boundary of the resulting surface is the knot 7₆. It gives rise to a 4 by 4 linking matrix, but in the end there are enough zeros that the determinant of LK-tLK^T is relatively easy to work out by hand. The result can be checked against the Knot Atlas, for example, to see that we end up with the right Alexander polynomial.
Assignment 6 (linked here) is due Friday March 31 by 11:59pm.

Unit 7: Braids
The relevant section from Adams is 5.4.
Lecture 31 (March 27)
From links to braids
Today we changed focus from knots and links to braids: whenever a link diagram winds monotonically around a fixed axis, it can be cut into a braid.

Lecture 32 (March 29)
The braid group: Guest lecture by Pamela Shah
Continued the discussion and saw that this new class of objects forms a group.

Lecture 33 (March 31)
Markov's theorem: Guest lecture by Pamela Shah
A theorem of Alexander states that every knot can be represented as the closure of a braid. This leads to the question of when two braids, possibly distinct, give rise to the same knot or link when closed. The answer is given by the Markov moves, which tell us how to relate braids with equivalent closures.
Assignment 7 (linked here) is due Thursday April 13 by 11:59pm.

Topics in Geometry: Introduction to Knot Theory

Math 309

Winter 2023

Homework assignments

Course Units