Math W4052: Knot Theory

Spring 2011

Taught by: Robert Lipshitz

Syllabus	Problem sets	Policies

(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):

• Knots and Links by Peter Cromwell.
• An Introduction to Knot Theory by W.B. Raymond Lickorish.

Some other useful books (also on reserve):

• The Knot Book by Colin Adams.
• Knot Theory by Charles Livingston.
• Knots and Links by Rolfsen.

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

• Khovanov Homology:
  • "On Khovanov's categorification of the Jones polynomial" by Dror Bar-Natan. Algebr. Geom. Topol. 2 (2002), 337–370 (electronic).
    • Journal article (AGT).
    • On the arXiv.
    • MathSciNet review.
    • My summary: The gentlest introduction to Khovanov homology that I know of.
  • "A categorification of the Jones polynomial" by Mikhail Khovanov. Duke Math J. 101 (2000), no. 3, 359-426.
    • Journal article.
    • On the arXiv.
    • MathSciNet review.
    • My summary: The original article introducing Khovanov homology. Recommended.
  • "Khovanov's homology for tangles and cobordisms" by Dror Bar-Natan. Geom. Topol. 9 (2005), 1443–1499.
    • Journal article.
    • On the arXiv.
    • MathSciNet review.
    • My summary: A more geometric perspective on Khovanov homology, which some people find clearer and some do not. Has led to a number of advances.
  • "Khovanov homology and the slice genus" by Jacob Rasmussen. Invent. Math. 182 (2010), no. 2, 419–447.
    • Journal article.
    • On the arXiv.
    • MathSciNet review.
    • My summary: The first combinatorial proof of the Milnor conjecture. Well written, and surprisingly easy (a posteriori). (The first proof, using gauge theory, is due to Kronheimer and Mrowka, in 1993.)
• Knot Floer Homology:
  • "A combinatorial description of knot Floer homology" by Ciprian Manolescu, Peter Ozsváth and Sucharit Sarkar. Ann. of Math. (2) 169 (2009), no. 2, 633–660.
    • Journal article.
    • On the arXiv.
    • MathSciNet review.
    • My summary: As the title says, this article gives a way of computing an existing invariant. So, it sometimes assumes you know about the invariant being computed. Still, if you don't worry about too much about the parts that you don't understand, it should basically make sense.
  • "On combinatorial link Floer homology" by Ciprian Manolescu, Peter Ozsváth, Zoltán Szabó and Dylan Thurston. Geom. Topol. 11 (2007), 2339–2412.
    • Journal article.
    • On the arXiv.
    • MathSciNet review.
    • My summary: This paper turns the combinatorial algorithm from above into a definition, and enhances it in a various directions. In principle, the paper is self-contained, but is written in an encyclopedic way that may make reading it (without some prior knowledge of the field) fairly hard work.

Prerequisites.

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:

• To further develop facility with the tools introduced in the first semester of topology, particularly the fundamental group, covering spaces, and cut-and-paste topology.
• To develop enough background to understand some of the topology research of current Columbia faculty, and how it fits into the narrative of mathematical progress.

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.

Policies.

Grading

Homework	40%
Midterm exam	30%
Final paper	30%

The lowest homework score will be dropped.

Homework

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.
03/09	Midterm exam
03/14-03/18	Spring break
03/21	The Alexander polynomial, II. In terms of covers.	L: pp. 54--64, 70--77.
03/23	The Alexander polynomial, II.V. More on covers.	(same)
03/28	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.
04/06	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.

• Problem Set 1. Due January 26.
• Problem Set 2. Due January 31.
• Problem Set 3. Due February 7.
• Problem Set 4. Due February 16.
• Problem Set 5. Due February 23.
• Problem Set 6. Due February 28.
• Problem Set 7. Due March 7.
• Problem Set 8. Due March 28.
• Problem Set 9. Due April 11.
• Problem Set 10. Due April 25.

Course materials for Math W4052, Spring 2011 by Robert Lipshitz are licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.

For questions or comments, please contact Robert Lipshitz.