Course: Honors topics in mathematics, Math 428
Time: 9:30--10:50 TR
Location: 341 Altgeld Hall
Instructor: Prof. Leininger
Email: clein (at) math.uiuc.edu
Office: 366 Altgeld Hall
Office hours (my office): TBA
Text: There is no required text for the class, though we may sometimes refer to the following:
- Colin Adams, The Knot Book
- Dale Rolfsen, Knots and Links
About the course: This course provides an introduction to knot theory and geometric topology. We will focus on classical knot theory of ``knotted circles in 3-space''. We will define a variety of invariants used to distinguish ``different'' knots. These invariants come from abstract algebra in the form of polynomials, groups and rings, and are derived by combinatorial and topological means. Along the way we will discuss some basic topology and the classification of surfaces. Grades will be based on class attendence/participation and homework.
Prerequisites: a certain level of mathematical maturity, including the ability to write and follow proofs is necessary. Abstract algebra would be helpful, but I will review the relevant material.
Class by the week:
- Week 1. Lecture 1, Lecture 2. The first three homework exercises are in these notes. One was not mentioned in class, namely...
Exercise I.3. Prove that there are only a countably infinite number of equivalence classes of knots in R3.
- Week 2. Lecture 3, Lecture 4. We discussed the first 5 exercises
at the beginning of class Thursday. There are more exercises in the
Note: A typo was pointed out in exercise I.8 in the notes: "sum to
1" should have said "sum to 0".
- Week 3. Lecture 5, Lecture 6. We discussed the exercises from last
week at the beginning of class Tuesday. If you missed because of the
weather, and would like a recap, feel free to drop by before class on
Note: Another typo spotted (missing negatives): Exercise II.3, you
should prove lk(J,K) = lk(K,J) = -lk(-J,K) = -lk(J,-K).
- Week 4. Lecture 7. Thursday of this week (2/10) will be devoted to presenting homework problems.
- Week 5. Lecture 8 (there is an exercise 15 here which is scratched out.. it's obvious, so skip it if you like). Lecture 9.
- Week 6. We mostly worked exercises Tuesday. After that, we started a new section on the link group and topology. Lecture 10. Thursday we began a section on topology. Exercises III.1--III.6 are to be turned in next Thursday. Lecture 11. Here are some notes on the problems: 29 topologies on a 3 point set, ε-δ implies continuity.
- Fixes: In Lecture 11, I forgot to fix the expresssion for the
``Manhattan metric'': as mentioned in class, the expression is different
when x = x': you should
then get |y-y'|. Another point to make is that this is
not what is usually called the ``Manhattan metric''... oh well, we'll call
- Week 7. Lecture 12. There is a minor
discrepancy with the theorem numbering here because I changed the order in
class (10, 11 and 12 were cyclically permuted). The exercises in Lecture
12 are due next Tuesday (March 8). For exercise 7, you only need to hand
in the proofs of parts 3 and 4 (though you should prove each part for
yourself). Here is the proof for part 3. Lecture 13. In the exercise on
stereographic projection, it's probably easier to write down the inverse
π-1:Rn → Sn, and write
it down as a map π-1:Rn →
Rn+1 (so you should have n+1 functions of n variables).
- Week 8. Lecture 14. Lecture 15. The problem(s) here are not to be turned in. We will discuss them in class later.
- Week 9. Lecture 16 and Lecture 17.
- Week 10 (3/29,3/31). Circumstances beyond my control have forced me
to be away this week. Professor Dunfield will be filling in for me on Tuesday:
please make sure you show up on time for class. There will be no class on
Thursday. Here are Profesor
Dunfield's lecture (much better pictures than mine!). Make sure
you've read through these, including the parts he didn't get to. We
will pick up from here next week.
- Week 11. Lecture 18 and Lecture 19.
- Week 12. Lecture 20 and
- Week 13. Lecture 22 and Lecture 23.
- Week 14. Tuesday we went over exercises. Lecture 24.
- Week 15. Lecture 25, the last lecture. Hope everyone enjoyed the semester and learned something interesting. If you want to talk about anything we did, or learn more about anything this semester, let me know and I'd be happy talk about it, or to point you in the right direction.
- The final exam is here. It's just for fun, but give it a try...
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