Spring Semester 2009

 

Course description: Most people think of mathematics as a computational discipline. In fact, that is only one small part of mathematics. In this course we will study the very non-computational subject of knot theory. We are all familiar with knots—square knots, granny knots, bowlines, etc. A mathematical knot is what we would obtain by gluing together the two free ends of one of these knots. That is, we can think of a knot as a continuous loop of string. The fundamental question is: how can we tell when two knots are the same and when they are different? In particular, is a given loop of string knotted at all? In this course we will develop tools that help us distinguish knots. Topics include colorability, the crossing number, the bridge number, the unknotting number, Seifert surfaces, genus, alternating knots, tangles, braids, torus knots, tangles, the Jones polynomial, the Alexander polynomial, and the HOMFLY polynomial.

This will not be a typical mathematics class. There will be no exams (evaluation will be done by means of homework, class participation, short quizzes, and a final project). The course will be run more as a seminar than as a lecture class. I expect you to read the material before class. That way we can focus more on questions that come up during the reading. You will present your work throughout the semester, culminating in a final project and presentation at the end of the semester. 

The tentative schedule is to cover sections 1.1-1.5, 2.1-2.3, 3.1-3.3, 4.1-4.3, 5.1, 5.4, 6.1-6.4 in the textbook, plus additional topics as time permits.

Grading: Your grade will be computed as follows.

• Homework, pet knot, quizzes (60%)  
• Research project (25%)
  
• Class participation: daily presentations, peer review of wiki pages, comments and questions in class, daily postings to the forum, attendance, and math/cs chat and EPaDel conference attendance (15%)

Exams: There will be no exams in this course.

Homework: Homework will be assigned and collected each class period. The assignments will be posted on the class wiki on the day of the class. Homework will be due at the beginning of the class. Late homework will not be accepted unless you obtained permission from me in advance. You are free to (and encouraged to) work together on the homework but you must turn in your own work and it should be written in your own words. Copying the work of a classmate is unacceptable. Each assignment will be graded on a 20-point scale. I will grade a randomly-chosen selection of the problems. At the end of the semester I will drop the two lowest homework grades.

Daily presentations: Each day one or more students will present homework problems at the board.

Forum: Each day you must make two posts on the class forum (you must do so before 8:00 on the day of class). In particular, you must answer the following questions:

Pet knot: Each student will pick one knot on the first day of class to be her or his "pet knot." Throughout the course we will return to the pet knots and apply the newly-acquired techniques to them. You will have your own wiki page devoted to your knot.

Research project: At the end of the semester each student will complete a research project. This will include a written document (on the wiki) and a presentation (during the last week of classes). More details will follow.

Wiki: Your work on the wiki will be graded on the following:

1. the mathematical correctness of the content,

2. the clarity of the writing,

3. the basics: grammar, spelling, punctuation, etc., and

4. the visual appearance (this includes the quality of the pictures, the use of latex, etc.).

Peer review of wiki: Each student is expected to give feedback to classmates about their work on the wiki. Exact details will follow.

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