University of Sydney / School of Mathematics and Statistics / Computational Algebra Group
Quantum Groups and Knot Theory
 

Organization

The course will take place on fifteen Wednesday mornings in the Fall of 2007, starting 12 September. The venue is
  • Until Oct 31st: Utrecht, University campus De Uithof, Buys Ballot Laboratory, room 105b.
  • After that, until Dec 19th: room 611AB of the Mathematics building (Budapestlaan 6)
The first three mornings will be dedicated to knot theory. If needed, a fourth meeting later in the course will be devoted to knot theory.

Typically, there will be two hours of lectures on each Wednesday, and one hour devoted to homework problems, each hour lasting 45 minutes, with intermissions in between. The responsibility for the homework problem sessions lies with Dan Roozemond.

Schedule

The schedule for the first three meetings:
  1. September 12: Introduction to knot theory: knot, link, knot diagram, orientation, Gauss code, Reidemeister moves, Seifert circles, Seifert surface, link invariant, crossing number, 3-coloring of a knot diagram, prime knot
  2. September 19: Second approach to knot theory: braid, Markov moves, fundamental group, Alexander polynomial, Conway polynomial, Vogel algorithm
  3. September 26: Braid group, writhe, skein relations, Kauffman algebra, Kauffman bracket, Markov trace, Jones polynomial

Examination

You will receive a separate grade for the first three meetings. For this part, the following rules apply for students who wish to get credit for the course:
  • Every week approximately ten homework problems will be handed out and published on the website.
  • For each of these problems it is mentioned how many points it is worth.
  • Each week, you can obtain at most four points.
  • Students may make their own selection of exercises. The perfect score of four points can be obtained in various ways: For example, doing four exercises worth one point each perfectly gets you the perfect score, as does doing three exercises worth two points each perfectly, as does doing eight exercises worth one point each only so-so. On the other hand, handing in only one homework challenge (worth three points) only so-so will get you about two points.
  • Students should hand in solutions either by e-mail, before the next Wednesday, 10.15am, or in person at the start of next week's lecture.
  • Cooperation between students is allowed (and indeed encouraged), but each student should write down the solution in their own words; solutions that are verbally identical are not acceptable.
  • There will not be a final examination. The students' grades are determined by their performance on the homework.