(There is also a web page on Courseworks.)

Time: MW 4:10-5:25
Place: Math 507
Instructor office hours: MW 5:30-6:30 in Math 424.
Teaching assistant: Kristen Hendricks
TA office Hours: M 10:00-12:00; Th 2:00-3:00 (in help room)

Textbooks (on reserve):

Some other useful books (also on reserve):

Papers for the last third of the semester. (This is in no sense an exhaustive list.)


Math W4051 (Topology).

Description and goals.

With questions that can be explained to a five-year old child but tools for answering them drawing on areas ranging from differential geometry to representation theory, knot theory rivals number theory as the jewel in the crown of mathematics. In this class, a follow-up to the topology course, we will start to explore these questions and connections.

In addition to learning some beautiful mathematics, the course's goals include:

Course structure.

The first half of the semester will be spent on basic background on knot theory and algebraic topology. Two key goals of the first half of the semester are to understand several definitions of the Alexander polynomial and the Jones polynomial. In the second half of the semester, we will survey some more advanced topics, including 3-manifold topology, knot Floer homology and Khovanov homology.

Knot theory is a broad subject, with many accessible topics we will not have time to cover. Instead of a final exam there will be a final paper, about a topic in knot theory not covered in the lectures and problem sets. In the second half of the semester, the problem sets will thin out, to leave more time for working on these papers. During the final exam period, we will probably have short presentations on the topics from the papers.



Homework 40%
Midterm exam 30%
Final paper 30%

The lowest homework score will be dropped.


Problem sets are due on Mondays at the beginning of class, except as noted below. If you can't make it to class, put it in my mailbox before class.

You're welcome to work on the homework together. However, you must write up your final answers by yourself. I consider writing them up together cheating.

You are also generally welcome to use any resources you like to solve the problems. However, any resource you use other than the textbook (Munkres) must be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including the help room and your classmates). Failure to cite sources constitutes academic misconduct.

Students with disabilities

Students with disabilities requiring special accommodation should contact Office of Disability Services (ODS) promptly to discuss appropriate arrangements.

Missed exams

If you have a conflict with the midterm exam date, you must contact me ahead of time so we can make arrangements. (At least a week ahead is preferable.) If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can.

Syllabus and schedule.

Note: C means Cromwell, L means Lickorish.

Date Material Textbook Announcements
01/19 Introduction to knot theory. Knots and links. Isotopy and ambient isotopy. Tame and wild knots. Examples.

C: §1.1--1.11, 2.10--2.12.

Welcome to W4052.
01/24 Reidemeister moves. Nontriviality of trefoil via 3-colorability.

C: §3.1--3.3, 3.8--3.9,
L: pp. 1--2.

01/26 Linking number. Nontriviality of Hopf link. The knot group. Nontriviality of trefoil and Hopf link via the knot group. L: pp. 3--11. Problem set 1 due.
01/31 Knot genus and unique factorization, I C: §5.1--5.7,
L: Ch. 2.
Problem set 2 due.
02/02 Knot genus and unique factorization, II (same)  
02/07 Simplicial homology I: the definition. C: §6.1--6.4. Problem set 3 due.
02/09 Simplicial homology II: key properties.    
02/14 More simplicial homology   Problem set 4 due.
02/16 Still more simplicial homology    
02/21 The Seifert matrix C: §6.5--6.6,
L: pp. 49--53.

Problem set 5 due.

[Drop deadline 02/22]

02/23 The signature and determinant. The unknotting number. C: §6.7, 6.8.  
02/28 The Alexander polynomial, I. Defintion, first examples. A genus bound. C: §7.1, 7.2.
Problem set 6 due.
03/02 Review of covering spaces. The infinite cyclic cover of a knot complement. L: pp. 66--69.  
03/07 Review. Some motivating problems in knot theory.   Problem set 7 due.


Midterm exam    
03/14-03/18 Spring break

The Alexander polynomial, II. In terms of covers.

L: pp. 54--64, 70--77.


The Alexander polynomial, II.V. More on covers.



The Alexander polynomial, IV. Skein formula.

C: §7.3--7.5,
L: pp. 79--84.
Problem set 8 due.
03/30 Knots and 3-manifolds: branched covers and surgery. L: §12  
04/04 The Jones polynomial C: §9.1--9.3,
L: Ch. 3.

Final paper topic due.


The HOMFLY-PT polynomial

C: §10.1--10.3,
L: Ch. 15.
04/11 Khovanov homology, I. Definition.   Problem set 9 due.
04/13 Khovanov homology, II. First examples.    
04/18 Khovanov homology, III. Key properties.   Paper outline due.
04/20 Khovanov homology, IV. The s-invariant and the Milnor conjecture.    
04/25 Knot Floer homology, I. Definitions and first examples.   Problem set 10 due.
04/27 Knot Floer homology, II. Properties and applications.    
05/02 TBA   Half paper due.


Problem sets.