Textbook: The knot book, by Colin Adams.

Materials: You should have some medium for tying knots. A rope or shoelace (about 2 feet long) should work. An extension cord is even better, because you can join the ends together.

Course goals: We will cover most of chapters 1–6 of the textbook. We will begin by introducing the notion of mathematical knots and links. The main topic in the course is developing methods to determine when two knots are actually distinct. Our main technique will be to introduce various invariants of knots and links including tri-colorability, unknotting number, bridge number, crossing number, genus, and various knot polynomials. Along the way, we will have an opportunity to stop and look at surface topology, three- dimensional topology, and possibly hyperbolic geometry.

Why knot theory? The theme of Math 4096 changes every semester. Why did I choose knot theory?

There are several good reasons. First, the subject is very accessible (as I hope you will agree when you start to read the book). You can get to a lot of cutting-edge knowledge very quickly. Second, the topic is current. We will be able to look at a several new discoveries that occurred in the 12 years since the textbook was written. Hopefully, this will let you get close to a living, breathing area of mathematical research.

Third, the subject is beautiful and visual. We will do a lot of visualization with ropes and reasoning with pictures. Nevertheless, it is quite rigorous: we will learn how to transform in- tuition about pictures into airtight proofs. This makes knots a great playground for practicing both reasoning skills and writing skills.

• Proofs in the homework. Homework assignments will mostly consist of proofs. They will be graded both on correctness and on clarity of exposition.
• Research paper. As part of a team of 2-3 students, you will need to complete an expository paper on a current topic related to the theme of the course. I will help you select topics (much more information is forthcoming). Your group will need to hand in a preliminary draft (for comments, discussion, and revision) and a final draft.
• In-class presentation. Your team will also give a 15-minute in-class presentation on the topic of your paper. Presenting difficult material orally is just as critical as presenting it in a written form, and involves somewhat different emphases.

• You should try to do all of the problems on your own before getting together with others. It does not benefit you (on exams and in the real world when you need to use math) to simply get solutions from your classmates! In fact, there is research suggesting that group work is much more productive when everyone has thought about the problems before getting together.
• Everyone must turn in their own solutions. In other words, you should write up your final solutions in the privacy of your own room (or your own library, cafe, bar, roof, etc.).

Here are a few guidelines for how to write up the proofs:

• Write up the problems in order, using only one side of the page and leaving lots of space for me to write comments. Please staple your sheets together.
  
• Begin each problem with a statement of that problem.
  
• Proofs should be written in complete sentences, with appropriate use made of mathematical notation (your textbook will serve as a guide to how to do this). Proofread what you've done to be sure that it's complete and makes sense. Remember that proof-writing is above all an act of communication, and that the ultimage goal is clarity.
  
• If you leave a small gap in a proof that you're not able to fill in, note this down. I'll try to indicate how to fill it in my comments.
  
• Start early! This way, if you are stuck, you can still discuss the problem with other students, with Danielle, or with me.

• The first day of classes is Tuesday, August 30.
• The last day to drop/add (tuition refund available) is Monday, September 12.
• The last day to withdraw (no refund) is Tuesday, October 25.
• Thanksgiving break is the week of Monday, November 21.
• The last day of classes is Thursday, December 8.
• The final exam is Thursday, December 15.