Prof. Ilya Kofman |
Office: 607 Mathematics, phone: 854-3210
Email: ikofman@math.columbia.edu Web site: http://www.math.columbia.edu/~ikofman/ |
Course Time and Place: 2:40pm - 3:55pm Tuesday and Thursday, 520 Mathematics Building
Text: Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology by Prasolov and Sossinsky
Supplementary Texts: The Knot Book by Adams and Knot Theory & Its Applications by Murasugi
Requirements: We will assume familiarity with linear algebra and some group theory. Basic Topology is not required, though it provides a useful perspective for knot theory. You will be asked to think about mathematical ideas and to prove what you think is true, so a certain amount of mathematical maturity is expected.
Material Covered: Basics of knots, links, and their diagrams, elementary knot invariants, Jones-type polynomials, Vassiliev invariants, braids, Seifert matrices and Alexander polynomial, 3-manifold topology, including Heegaard diagrams, surgery, and branched covers of knots.
Homework: Assignments will be announced in class and then posted on this website. Any changes will be announced in class. Incomplete work with good progress will be rewarded. I highly recommend working jointly on homework problems with fellow students, but in the end you must hand in your own work.
Grading: The course grade is apportioned as follows: homework 40%, take-home midterm exam 30%, final in-class presentation and written report 30%.
Help: My office hours are right after class in my office 607 Mathematics. The help room, 406 Mathematics, is open with people to answer questions all day long on weekdays, but some topics we cover require specialized knowledge.
Optimal Method of Study: (1.) Come to class. (2.) Read the relevant sections after class. (3.) Do the homework. Leave time to think--do not put homework off until it is due! (4.) Compare your solutions with other students to improve what you hand in. (5.) Come to office hours or the help room with any remaining questions.
Goals: The primary goal of this course is to introduce you to knot theory, which touches upon many branches of mathematics. As for any topics course, an essential goal is to learn how to do research mathematics, including how to write concise but complete proofs, and how to present to others what you have learned.
The schedule below is tentative and I am open to suggestions for other topics.
Week | Topic | Reading | Homework | Due |
Sep. 2 | Elementary knot invariants | Ch. I, Adams ch.1,3 | I.1: 1,2,4,5, Adams: 1.6, 1.7 | Sep. 9 |
Sep. 9 | Jones polynomial, Kauffman bracket, alternating knots | Ch. II.3 | II.3: 1,2,3,4,5 | Sep. 16 |
Sep. 16 | Spanning tree expansion for Jones polynomial | class notes | assigned in class | Sep. 23 |
Sep. 23 | Generalized knot polynomials | Adams ch.6 | none | |
Sep. 23 | Vassiliev invariants | Ch. II.4, Murasugi ch.15 | II.4: 2,3,4, Murasugi: 15.3.2, 15.3.4, 15.3.5, [BONUS: 15.3.1(1)] | Sep. 30 |
Sep. 30 | Braids | Ch. III | 5.1, 6.1, 6.2, 6.4 | Oct. 9 |
Oct. 7 | Braids and Seifert surfaces | Adams ch.4, Murasugi ch.5 | Adams: 4.14, 4.16, 4.18 - 4.22 | Oct. 14 |
Oct. 14 | Seifert matrices and Alexander polynomial | Murasugi ch.5, ch.6 | Murasugi: 4.5.5, 5.3.1, 5.3.2, 5.4.1 | |
Oct. 21 | Signature and knot cobordism, Heegaard diagrams for 3-manifolds | Ch. IV | Take-home Midterm Exam | |
Oct. 28 | Lens spaces and surface homeomorphisms | Ch. IV,V | ||
Nov. 6 | Surgery of 3-manifolds | Ch. V,VI | ||
Nov. 11 | Kirby calculus | Ch. VI | ||
Nov. 18 | Branched coverings | Ch. VII | ||
Nov. 25 | Skein invariants of 3-manifolds | Ch. VIII | ||
Dec. 2 | Your choice | Presentations | In-class presentation and written report |