Topics in Geometry: Introduction to Knot Theory

Math 309

Fall 2020

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?"

The course will meet using Zoom, scheduled in Canvas.

The course is undergoing an adaptation to a fully-online version in response to the Covid 19 pandemic; a previous version of this course can be found here. Please bear with us!

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 this term, 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.

• Instructor
  Liam Watson
  liam(at)math(dot)ubc(dot)ca
  Office hours Mondays 2:00 to 3:00 and Thursdays 11:00 to 12:00. Meetings are scheduled in Canvas.

• Teaching assistant
  Mihai Marian.
  mihmar(at)math(dot)ubc(dot)ca
  Office hours Wednesdays 12:00 to 1:00 and Fridays 1:00 to 2:00.

• Where, when...
  The class meets Tuesdays and Thursday from 9:30 to 11:00 using Zoom. (To avoid screen exhaustion, we won't use the full 1.5 hours.) Meetings are scheduled in Canvas.

• Evaluation
  Your final grade will be calculated based on your submitted homework assignments. Please note that no late assignments will be accepted. Group work is strongly encouraged, but it is an absolute requirement that you submit your own work for grading. As you are aware, copying solutions from another student, from the web or from any other source, and turning them in as your own, is a violation of the Academic Code.

• Reference
  The Knot Book by Collin Adams, available at the UBC bookstore.

• Other reading
  
  The book Knots and Links by Dale Rolfsen is a homegrown UBC classic, and is worth a look, but it assumes more topology background than we will in this course.
  
  Dror Bar-Natan's incredible Knot Atlas.

              Homework assignments

Please note that no late assignments will be accepted. 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 this term, 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.

• Assignment 0
  
  This is an essential test, consisting of two pages, due on Monday September 21 2020 by 11:59pm.
  
  Instructions: page 1 should consist of your name and student number together with the assertion that you have read and understood the four guidelines above, and page two should include a drawing of a knot. Please submit your work according to the instructions above.
• Assignment 1 (linked here) is due on Friday September 25 by 11:59pm.
• Assignment 2 (linked here) is due on Monday October 5 by 11:59pm.
• Assignment 3 (linked here) is due on Wednesday October 14 by 11:59pm.
• Assignment 4 (linked here) is due on Wednesday October 28 by 11:59pm.
• Assignment 5 (linked here) is due on Monday November 9 by 11:59pm.
• Assignment 6 (linked here) is due on Monday November 23 by 11:59pm.
• Assignment 7 (linked here) is due on Monday December 7 by 11:59pm.

              Course Units

• Unit 1: Foundations
  The relevant sections from Adams are 1.1 through 1.5.
• Lecture 0 (September 8)
  Prelude: parametrizing curves in the torus
  As a point of entry for studying knots, we reviewed some material from calculus about parametrized curves, and looked specifically at some parametrized curves that wind around in a torus. Key components will be reviewed next time, but due to the overlap with imagine day activities, lecture notes have been posted here.
  
  Lecture 1 (September 15)
  Knot diagrams
  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 there are between diagrams that represent the same knot.
  
  Lecture 2 (September 17)
  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 3 (September 22)
  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 Friday September 25 by 11:59pm.

• Unit 2: Linking and knotting
  The relevant sections from Adams are 1.4, 3.1, 3.3, and 5.1.
• Lecture 4 (September 24)
  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 5 (September 29)
  Unknotting numbers
  We considered two natural but very difficult to compute numerical invariants of knots: the crossing number and 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 6 (October 1)
  Torus knots
  A large supply of non-alternating knots are provided by torus knots, which we encountered in earlier lectures. I gave 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 Monday October 5 by 11:59pm.

• Unit 3: Knots and continued fractions
  The relevant sections from Adams are 2.2, 3.1, and 3.2.
• Lecture 7 (October 6)
  Tangles
  Tangles provide the building blocks for knots that Conway used for tabulation. This led us to consider 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.
  
  Lecture 8 (October 8)
  Bridges
  We considered 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 9 (October 13)
  Rational knots
  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.
• Assignment 3 (linked here) is due Wednesday October 14 by 11:59pm.

• Unit 4: Polynomial invariants
  The relevant section from Adams is 6.1.
• Lecture 10 (October 15)
  From diagrams to polynomials
  Today we derived a divide-and-conquer strategy for converting a diagram into a Laurent polynomial with integer coefficients.
  
  Lecture 11 (October 20)
  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 12 (October 22)
  Trefoils
  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 Wednesday October 28 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 12 (October 27)
  Alternating diagrams
  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 13 (October 29)
  Alternating diagrams, continued
  Outstanding from last time was 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 14 (November 3)
  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 s 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 and will look at more closely next time. 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.
  
  Lecture 15 (November 5)
  The skein module
  Reviewing our discussion from last time, we went over the construction of the skein module (and the skein algebra) following the terminology introduced in Rolfsen's paper.
• Assignment 5 (linked here) is due Monday November 9 by 11:59pm.

• Unit 5: Surfaces and the Alexander polynomial
  The relevant sections from Adams are 4.1 through 4.3 as well as parts of 6.3.
  
  Lecture 16 (November 10)
  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 17 (November 12)
  Surfaces (lecture by Mihai Marian)
  
  Lecture 18 (November 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. This gives an assignment of surfaces to diagrams, and 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 19 (November 19)
  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!
• Assignment 6 (linked here) is due Monday November 23 by 11:59pm.

• Unit 6: Braids
  The relevant section from Adams is 5.4.
  
  Lecture 20 (November 24)
  The braid group
  Today we changed focus from knots and links to braids, and saw that this new class of objects forms a group.
  
  Lecture 21 (November 26)
  Markov's theorem
  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.
  
  Lecture 22 (December 1)
  Artin Combing
  The trivial knot is a hard thing to detect, but owing to the additional structure of braids the trivial braid is easily detectable. Artin showed how to do this using combing. From our point of view, this is a strategy that first considers the permutation associated with a braid, and then in the case where this permutation is trivial, considers twisting among some subset of the strands. For those of you with a background in (or those planning to take a course on) group theory, Artin's construction gives an excellent example of a semi-direct product of groups.
  
  Lecture 23 (December 3)
  Polynomials, revisited
  
• Assignment 7 is due Monday December 7 by 11:59pm.