m363.html

MATH 363 Knot Theory, Spring 2012 (CRN 9522)

Course Description: An introduction to the mathematical theory of knots and links, with emphasis on the modern combinatorial methods. Among specific topics to be covered are: Reidemeister moves on link projections, ambient and regular isotopies, linking number, tricolorability, rational tangles, braids, torus knots, seifert surfaces and genus, the knot polynomials (bracket, X, Jones, Alexander, HOMFLY), crossing numbers of alternating knots and amphicheirality. Connections to theoretical physics, molecular biology, and other scientific applications will be pursued in term projects, as appropriate to the background and interests of the students. Prereq: MATH223

Instructor: Joel Langer, Yost 310, phone: 368-2897, email: joel.langer@case.edu

Class schedule: TuTh 1:15--2:30 pm, Sears 374.

Text: The main text for the course will be The Knot Book, by Colin Adams, W.H. Freeman and Co. Some other (mostly more advanced) references include:
An Introduction to Knot Theory, by W. B. Raymond Lickorish, Springer-Verlag; Knot Theory and its Applications, by Kunio Murasugi, Birkhauser;
Knots and Physics and Knots and Applications, written/edited by Louis Kauffman and published by World Scientific.

Web Resources: There are web sites related to all aspects of knots, including mathematics and theoretical physics (expository articles and course materials as well as current research), molecular biology, computer graphics (not to mention art, magic, practical knots and applications). Though it is no longer updated regularly, the first site on the following list is still an excellent place to get an overview (and to get ideas for a term project -- see explanation below) :

Knots on the Web -- see especially Knot Theory and Knot Software -- by Peter Suber. Certainly one of the nicest and most comprehensive lists of knot-related web sites.
The KnotPlot site, by Rob Sharein. Though it has been around for some years now, KnotPlot appears to be the most advanced graphics package for knots, as evidenced by the many beautiful images and animations on this site.
books on knots listed by the CWRU library electronic catalogue.
search "knots" on Amazon.com
Journal of Knot Theory and its Ramifications
Morwen Thistlethwaite (homepage of a knot theorist).

The Term Project: The term project will provide an opportunity for students to explore some of the interesting connections between knot theory and other scientific fields, or to pursue a particular mathematical topic in greater depth. Groups of several students will settle on topics by the sixth week of the semester (in consultation with me). Members of a group will cooperate to develop their topic and give 45 minute presentations during the last two weeks of class (and the scheduled final exam period). Each member of the group will submit an individual written term paper, typically about 8-10 pages in length (excluding figures and supplementary materials). The project will be due on our last class day, Thursday, April 26.
(The grade for the term project will be based on the class presentation as well as the written paper.)

Course Requirements and Grading: The grade will be based on the two mid-term exams (25% each), on the term project (30%), and on class participation (20%). The class participation score will reflect attendance, in-class discussion of the weekly homework assignments, and the online homework journal, which students will take turns contributing to. The homework journal (as well as class notes) will be accessed through Blackboard (see blackboardcontents).

Joel Langer's home page