Undergraduate seminar on knot theory: MATH UN3952 (Spring 2020)
- Meetings: Tuesdays 5:40-7:25pm, Hamilton 503
- Section leader: Henry Liu (hliu at math dot columbia dot edu)
- Course coordinator: Daniele Alessandrini
The remainder of this seminar will be online via Zoom.
The goal of this seminar is to explore knot invariants, both modern and traditional. Knots and related subjects play a deep role in modern mathematical physics. Specific topics will be chosen based on participant interest, but the first few weeks will be an introduction to the general theory of knots. Our focus will be more conceptual and computational than technical.
Content aside, this seminar will also hopefully be a good training ground for how to give and how to learn from mathematical talks. Grades-wise, participation and good communication will be just as important as the mathematics.
Grading policy
Your grade will be based on participation as an audience member and as a speaker. Audience members are expected to support the speakers by asking good questions and giving constructive comments. Speakers are expected to give lucid, cohesive, and engaging presentations.
At the end of every meeting, please submit to me something (in any form) with your name and three big/interesting ideas you learned from each talk, in your own words. This will serve as an attendance record (for grading) and also as encouragement for you to synthesize what you are hearing from the talk.
Topics and references
Our main references for the fundamentals of knots will be the following books. They are listed in order of increasing mathematical sophistication.
- [A] C. Adams, The Knot Book
- [L1] C. Livingston, Knot Theory
- [L2] W. B. R. Lickorish, An Introduction to Knot Theory
Good, pedagogical introductions to the basics of knot theory are: chapters 1-6 of [A], chapters 2-4 of [L1], and chapters 1-2 of [L2].
The Jones polynomial and related polynomial invariants are discussed in: chapter 6 of [A], chapter 10 of [L1], and chapters 3, 6, and 8 of [L2]. Different perspectives on the Jones polynomial are introduced well in part I of [K]:
- [K] L. Kauffman, Knots and Physics
Here are some interesting topics for talks. Those of you who have not signed up for a topic yet, please pick from one of the following or, even better, suggest your own. For most topics, I have listed some suggested references; feel free to use other sources.
- Temperley-Lieb algebra and the original definition of the Jones polynomial ([K] § 7, [L2] §§ 13-14)
- Knot polynomials as quantum group invariants ([K] §§ I.8-11)
- Knot/link homology (categorification)
- Vassiliev (finite type) invariants
- Anything we haven't covered in [A], [L1], [L2], and [K]. For one-person topics, [K] is especially good.
Schedule
Each individual talk should take approximately 50 minutes. Email me a title, a short description, and any relevant references for your talk a few days before the date of your talk.
If you want to sign up for, confirm, or cancel a talk for a particular date/topic, please ensure that I am aware of the change at least one week in advance.
Thurs Jan 30 | Organizational meeting |
Tues Feb 04 |
Alaedine Belhouari and Nouhaila El Majidi An Introduction References: [A] §§ 1-2 and [L1] §§ 1-2. |
Tues Feb 11 |
Alaedine Belhouari and Nouhaila El Majidi An Introduction (cont'd) References: [A] §§ 1-2 and [L1] §§ 1-2. Julian Christensen Representations of Knots References: [A] §§ 2.2, 2.4, 4.1 |
Tues Feb 18 |
Daniel Ortega-Venni Characterizing Knots Through Numbers and Colors References: [A] §§ 2.3 and 3, [L1] § 3.2 Sara Bousleiman and Yuri Hayashi References: [A] §§ 3-4 and [L1] §§ 3-4, 7 |
Tues Feb 25 |
Sara Bousleiman and Yuri Hayashi References: [A] §§ 3-4 and [L1] §§ 3-4, 7 |
Tues Mar 03 |
Abdoulaye Diallo and Rohan Sukhdeo References: [A] §§ 4.3, 6.1 |
Tues Mar 10 | Class canceled (university-wide) |
Tues Mar 17 | Spring break |
Tues Mar 24 | Class canceled (university-wide) |
Tues Mar 31 |
Cynthia Mao References: [A] § 6, [L1] § 10, [L2] § 3, [K] §§ I.1-5 |
Tues Apr 07 |
Elias Romero Colors, Brackets, Tangles, and Nature References: [A] §§ 7.4-8.3, [K] §§ II.7-8 Ryan Eppolito The concordance group References: Arunima Ray, Knots, four dimensions, and fractals |
Tues Apr 14 |
Larry Hong and Joon Young Kang References: [A] § 7.1 |
Tues Apr 21 |
Zeina Laban Knot Theory in Quantum Cryptography References: Marzuoli and Palumbo, Post Quantum Cryptography from Mutant Prime Knots Ben Brimacombe Deciding Knot References: Koenig and Tsvietkova, NP-hard problems naturally arising in knot theory |
Tues Apr 28 |
Aaron Zheng Types of Knots References: [A] § 5 Gal Polani Title: TBA References: [K] §§ I.8-11 |