Columbia University W4052 section 001
Introduction to Knot Theory

Basic information

Call number: 93650
Room/Time: MW 10:35am--11:50am, 307 Math
Instructor: Mikhail Khovanov
Office: 620 Math
Office hours: MW 1:30-2:30 or by appointment
E-mail: khovanov@math.columbia.edu
Teaching assistant: Krzysztof Putyra, putyra@math.columbia.edu
TA office hours: Tuesday 1:00-2:30pm, room 610
Round table: Friday 5pm, room 528
 
Midterm: April 4
Webpage: www.math.columbia.edu/~khovanov/knottheory2012
 

Syllabus and resources

Syllabus:
Definitions of knots and links. Reidemeister moves. Linking number.
Knot colorings.
Operations on knots. Connected sum, mirror image.
Braids, Alexander theorem. Markov's theorem.
Kauffman bracket, Jones polynomial, applications. Alternating knots.
Tangles and Temperley-Lieb algebra.
Fundamental group of knot complement.
Classification of surfaces, Euler characteristic, orientation. First homology group.
Seifert surfaces of knots.
Seifert matrix, Alexander polynomial, signature.
Skein relation for the Alexander polynomial. Fox calculus.
HOMFLYPT polynomial.
Link homology and its applications.
Knots and 3-manifolds. Surgery.

The following books will be placed on reserve in the math library:

Knots and Links, by Peter Cromwell;
The Knot Book, by Colin Adams;
On knots, by Louis Kauffman;
Knot theory, by Charles Livingston;
An introduction to knot theory, by Raymond Lickorish.

An incomplete list of other books on knot theory:
Knots and links, by D.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 J.Birman; Knot theory, by V.Manturov.

Homework

will be assigned on Mondays, due Monday the following week before class.
The numerical grade for the course will be the following linear combination:
50% homework, 25% midterm, 25% 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      
Homework 9      
Homework 10      
Homework 11      

Practice quiz       Solutions

Online resources

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

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 classification.
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 Prerna 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