TuTh 2-3:30, 2 Evans
Office hours: T 1-2, W 12:30-2 and by appointment for project meetings, 1039 Evans
Textbooks: The Knot Book, Colin Adams (required); Topology from the Differentiable Viewpoint, John Milnor (recommended)
Syllabus with course requirements and important dates.
If you require DSP accommodations or are participating in an extracurricular which you expect to present scheduling conflicts, please contact the relevant offices and let me know as soon as possible.
The short and long projects are two of the three the cornersones of this class. As such, the course material in January-March is geared towards helping you make enough sense of the field of knot theory that you will be willing and able to choose a topic and investigate it in depth. This means we will be learning the basics of algebraic and differential topology, and then diving into knot invariants with an exploratory, example-focused mindset.
The first project will be short (3-6 pages, with in-class discussion, due second week of March) and the second long (7-10 pages, with 15 minute in-class presentation, due in the last week of class). Both must be typed in LaTeX (here to download the software to your computer, or here or here to use online compilers), so I suggest starting to use LaTeX to submit your homework as soon as possible. I'll meet with every student/team several times before they turn in their project. More details are in the syllabus.
Short project guidelines and suggestions (suggestions will be updated over time) and rubric. Long project rubric/guidelines and suggestions. From previous versions of this course: 2016, 2012 short project, 2012 long project.
Example long project: I was given permission by this student to post his final project, Braid Groups, Representations, and Algebras.
The other cornerstone of the course is to give you a piece of the map to the universe of research mathematics. As such, the lectures in April will all be grouped around interesting subfields related to knot theory.
Since this class relies so heavily on interaction between all of us, attendance is required. If you are unable to attend a lecture, you must let me know before hand and submit all required work on time. Because emergencies happen, each student can miss 2 classes which are not project presentations with no penalty to their grade. If you have an emergency which causes you to miss class, please let me know as soon as possible so that we can make sure you are caught up on the material.
It is important that you complete the assigned reading and homework before it is due. That way you can make the most of your opportunities to talk with me and your fellow classmates about the material. The reading, like the lectures, is intended to help you make your own map of the field of knot theory. Homework is intended to help you keep tabs on your understanding. It is NOT supposed to be difficult or voluminous, because the projects are the main focus of the course. However, in the case of homework or anything else related to the course, if you have any questions, please come talk to me during office hours. You are encouraged to discuss the homework with other students, however you must write it up on your own.
It's not necessary that you use LaTeX to submit your homework, however, it is good practice to learn by using it whenever it is reasonable. Because there is a lot of drawing in this class, feel free to use your own judgement to interpret what "reasonable" means for you -- if you want to learn to generate images with your computer, please feel free! Just know that there will be no penalty for turning in handwritten homework.
Supplementary resources will be linked to the appropriate dates below. If they aren't listed in the "Reading" column then they're not required.
| Date | Topic and supplemental resources | Reading | Homework due on this date (Thursdays only) |
|---|---|---|---|
| Tues Jan 16 | Course overview. Intro to knots and links. Knot presentations. | none | |
| Thurs Jan 18 | Knot presentations, linking number. REU discussion. | Adams sections 1.1, 1.3, 1.4 | none |
| Tues Jan 23 | Knot operations. Extra reference on knot concordance, the group related to knot composition, if you're interested. The knot table. Notes through today's lecture. | Adams section 1.2 | |
| Thurs Jan 25 | Tricolorability. Extra references on p-coloring if you are interested. Torus knots. | Play some torus games, plus Adams sections 1.5, 5.1 | Adams exercises 1.10, 1.17, 1.8 |
| Tues Jan 30 | More tricolorability and torus knots. | Adams section 5.1 | |
| Thurs Feb 1 | Satellites and cables, mutation. Twist knots. | Adams sections 5.2, 5.4. | Adams exercises 1.21, 1.29, plus the following: To switch the sign on one of the integers determining a torus knot, switch the direction which the knot travels with respect to either the meridian (changing the sign on p) or the longitude (changing the sign on q). When p and q are both positive, orient the meridian from top to bottom on the part that's in front of the torus (drawn solid, not dotted) and the longitude counterclockwise. What's the relationship between the (p,q), (-p,-q), and (p,-q) torus knots? |
| Tues Feb 6 | Braids. | Adams section 5.4. Extra references on braids. | |
| Thurs Feb 8 | The braid group. | Adams section 5.4, chapter 3 | Adams exercises 5.13, 2.23, 5.17 |
| Tues Feb 13 | Markov equivalence. Unknotting number. | Adams chapter 3 | |
| Thurs Feb 15 | Bridge number, crossing number. | Adams chapter 3 | Adams exercises 5.22, 5.26 |
| Tues Feb 20 | Surfaces 101. | Adams section 4.1 | |
| Thurs Feb 22 | Surfaces with boundary. | Adams section 4.2 | Adams exercises 3.5, 3.6, 3.13 |
| Tues Feb 27 | Seifert surfaces and knot genus. | Adams 4.3 | |
| Thurs Mar 1 | Fundamental group. Group presentations. | Hatcher section 1.1 (optional: skip proof that fundamental group of circle is Z) and optional extra reading Lickorish, "An Introduction to Knot Theory," Chapter 11 | Short project progress report due |
| Tues Mar 6 | Wirtinger presentation. Colorings revisited. | Wirtinger presentation, Wirtinger presentation and colorability, a history of coloring. | |
| Thurs Mar 8 | Continued from last time. | Adams section 6.1 | Adams exercises 4.7, 4.22. Find an explicit homotopy between the paths f(t)=(cos(2Pi*t),sin(2Pi*t)) and g(t)=(cos(4Pi*t)cos(2Pi*t),cos(4Pi*t)sin(2Pi*t)) in the plane R^2. |
| Tues Mar 13 | Continued from last time. Bracket and Jones polynomials. | Adams sections 6.1 | |
| Thurs Mar 15 | Short project discussion day | Bring your short projects! | |
| Tues Mar 20 | More Jones polynomial. Jones polynomials of alternating knots. | Adams section 6.2 | |
| Thurs Mar 22 | Alexander and HOMFLY polynomials. | Adams section 6.3. Extra references: Lickorish, An Introduction to Knot Theory, Ch 6, 15 and 16. For computing from a Seifert surface, this undergraduate thesis should be helpful. | Compute the fundamental group of either trefoil using the Wirtinger presentation. Find a homomorphism from the fundamental group of the complement of your trefoil to the dihedral group D_3 (in class I used the notation D_{2n} for the dihedral group of symmetries of a regular n-gon; the standard notation is D_n, so here I mean the group of symmetries of an equilateral triangle) which corresponds to a tricoloring of your trefoil as discussed in class. |
| Tues Apr 3 | Continued from last time. Simplicial complexes. | There are a lot of ways to construct simplicial homology (it's also equivalent to singular homology, which is what is usually used for proofs at the graduate level). Here are some good references for our purposes: Farb, Wilton, and of course Ch 2, section 1 of Hatcher. | Long project 1-1 meeting this week |
| Thurs Apr 5 | Simplicial homology continued. | see above references | Adams exercise 6.9, compute the Jones polynomial of the figure-8 knot, compute the Alexander polynomial of the Hopf link using (a) a resolving tree and (b) a Seifert surface |
| Tues Apr 10 | Euler characteristic. Alexander polynomial and genus. Mayer-Vietoris sequence. Start Khovanov homology. | On Khovanov homology. | |
| Thurs Apr 12 | More Khovanov homology. | ||
| Tues Apr 17 | More Khovanov homology. | Long project first draft due; long project 1-1 meeting this week | |
| Thurs Apr 19 | Computing from LES; start construction. | Bar-Natan's approach and Schumakovitch's approach. | |
| Tues Apr 24 | Finish construction and compute Kh(trefoil). Start project presentations. | ||
| Thurs Apr 26 | Long project presentations | ||
| Tues May 1 | Long project presentations | ||
| Thurs May 3 | Long project presentations | ||
| Mon May 7 | Long project papers due by 5PM |
Summer programs are a great way to continue your exploration of mathematics! Here are some links to information about summer math programs, some of which are REUs ("Research Experience for Undergraduates," run by the NSF) in pure math and plenty of which are not. These aren't exhaustive -- if you're interested, keep asking around and searching on your own, or please come talk to me! The Career Center is also a good source for summer opportunities. Keep in mind that many deadlines are coming up soon.
| NSF list |
| AMS list |
| More general list than the above two |
| Math StackExchange discussion |
| Bowdoin's advice for Summer 2017 (should still be useful) |
| MathPrograms.org |
Here are some drawing tools mentioned in class. TikZ is a package for drawing in LaTeX. Here's the instruction manual. Inkscape is another popular tool for drawing. If you're interested in mathematical software, Sage is free, and you can request a Mathematica license from the university. SnapPea and its Python-based extension, SnapPy, are tools for understanding certain types of three-dimensional spaces, which can also be used to study knots. The KLO software is intended to study three- and four-dimensional spaces and can also be used to study knots.