Welcome to the summer tutorial Knots and Primes. Chao and Charmaine are excited to share with you this piece of beautiful and surprising story of mathematics. Hope you have fun in exploring some analogy between knots and primes this summer!


Time: We will meet from July 2 (Mon) to August 11 (Fri), MWF 7pm-8pm. With the exception of the first week due to Independence Day: we will have no meeting on July 4 but will have two 1.5-hour meetings on July 2 and July 6, 7pm-8:30pm.

Location: Science Center 109.

Textbook: Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, Springer 2012. Each enrolled student has a $120 budget for books.

Recommended references on algebraic topology and number theory:

  • Algebraic Topology by Allen Hatcher (this is available electronically on Hatcher's website for free)
  • The Knot Book by Colin Adams
  • Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication by David Cox
  • A Course in Arithmetic by J.-P. Serre
  • Number Theory 1: Fermat's Dream, Number Theory 2: Introduction to Class Field Theory by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito

Check out the syllabus!

TopLecture Notes

Lecture Notes in one file (Updated 2022/01/03, thank Xiaoxiang Zhou for a list of typos)

No. Date Presenter Topic Lecture Notes
1 07/02/2012 Charmaine Sia Overview and Knot groups
2 07/06/2012 Chao Li Quadratic reciprocity
3 07/09/2012 Charmaine Sia Covering spaces
4 07/11/2012 Chao Li Number rings and etale coverings
5 07/13/2012 Charmaine Sia Linking numbers
6 07/16/2012 Chao Li Etale fundamental groups
7 07/18/2012 Chao Li More etale fundamental groups
8 07/20/2012 Chao Li p-adic integers and prime groups
9 07/23/2012 Charmaine Sia Alexander polynomials
10 07/25/2012 Charmaine Sia More Alexander polynomials
11 07/27/2012 Chao Li Class field theory
12 07/30/2012 Charmaine Sia Asymptotic formulas on homology groups
13 08/01/2012 Chao Li Ideal class groups
14 08/03/2012 Chao Li Iwasawa theory


Chao and Charmaine are glad to try our best to help you on the projects. Simply send us an email if you need any.

Date Presenter Topic Notes
08/06/2012 Allen Yuan Braid groups
08/08/2012 Michael Jemison Milnor invariants and multiple residue symbols
08/08/2012 Seth Neel Genus theory of binary quadratic forms
08/10/2012 Arpon Raksit Jones polynomial and Khovanov homology
08/10/2012 Jacob Postema Homology groups and class groups