Columbia University W4052 section 001
Introduction to Knot Theory

Basic information

Call number: 77319
Room/Time: Tu, Th 11:40am--12:55pm, 407 Math
Instructor: Mikhail Khovanov
Office: 620 Math
Office hours: Tuesday 4-5pm, Thursday 10:30-11:30am or by appointment
Midterm: Tuesday, March 12

Syllabus and resources

Definitions of knots and links. Reidemeister moves. Linking number.
Knot colorings.
Operations on knots. Connected sum, mirror image.
Fundamental group of knot complement.
Braids, Alexander theorem. Markov's theorem.
Kauffman bracket, Jones polynomial, applications. Alternating knots.
Tangles and Temperley-Lieb algebra.
Classification of surfaces, Euler characteristic, orientation. First homology group.
Quick review of homology and homotopy theories.
Seifert surface, Seifert matrix, Alexander polynomial, signature.
Manifolds and 3-manifolds. Surgery on knots and branched covers.
Link homology and its applications.
Knots and complexity theory.

Knots Knotes by Justin Roberts.
An introduction to knot theory, by Raymond Lickorish. Can be bought online.
For many free online resources see links below.


will be assigned on Thursdays, due Thursday the following week before class.
The numerical grade for the course will be the following linear combination:
50% homework, 20% midterm, 30% presentation at semester's end. The lowest homework score will be dropped.
Homework 1      
Homework 2      
Homework 3      
Homework 4      
Homework 5      
Homework 6      
Homework 7      
Homework 8      


Knot knotes, by Justin Roberts. Close to what we'll cover in the first half of the course.

Introduction to Hopf algebras, by Ken Brown.

The Trieste look at knot theory, by Jozef Przytycki. Introduction to knots and a survey of knot colorings.
3-coloring and other elementary invariants of knots, by Jozef Przytycki.

Knot theory, a book by Vassily Manturov.
Short review of braids, by Dale Rolfsen.
An Introduction to braid theory, by Maurice Chiodo.
An elementary introduction to the theory of braids, by Roger Fenn.

Catalan numbers, by Tom Davis.
Catalan addendum, by Richard Stanley.
About the Temperley-Lieb algebra: by V.S.Sunder, Anne Moore, Dana Emst.

Classification of surfaces, by Allison Gilmore. More surfaces, statement of clas sification.
A guide to the classification theorem for compact surfaces, by Jean Gallier and Dianna Xu.

Algebraic topology, by Alan Hatcher. Homology is covered in chapter 2.
An introduction to homology, by Pre rna Nadarthur.
Notes on homology theory, by Abubakr Muhammad.

An ABC of categories, by Tom Leinster.
Basic category theory, by Jaap van Oosten.
Category theory, by Steve Awodey.

Topological invariants of knots: three routes to the Alexander polynomial, by Edward Long.
Knot theory and the Alexander polynomial, by Reagin McNeill.

Data on knots and their invariants:
The Knot Atlas (wiki), by Dror Bar-Natan and Scott Morrison. Among other info, it contains Rolfsen's table of knots up to 10 crossings.
Table of Knot Invariants, by Charles Livingston and Jae Choon Cha.
The KnotPlot Site

Some knot theory books that you may find in the library:

Knots and Links, by Peter Cromwell;
The Knot Book, by Colin Adams;
On knots, by Louis Kauffman;
Knot theory, by Charles Livingston;
Knots and links, by Dale Rolfsen;
Introduction to knot theory, by R.Crowell and R.Fox;
Knot theory and its applications, by K.Murasugi;
A survey of knot theory, by A.Kawauchi;
Braids, links and mapping class groups, by Joan Birman.