Math 9023: Knot Theory and Low-Dimensional Topology
Fall 2018
| Meets: | Mon/Wed 9:00 - 10:20 AM in Wachman Hall, room 527 |
| Instructor: | David Futer |
| Office: | 1026 Wachman Hall |
| Office Hours: | Mon 10:30 - 12:00, Tue 2:30-4:00 PM |
| E-mail: | dfuter at temple.edu |
| Phone: | (215) 204-7854 |
Course content: This course will survey the modern theory of knots, coming at it from several very distinct points of view. We will start at the beginning with projection diagrams and the tabulation problem. We will proceed to several classical polynomial invariants, which can be constructed via the combinatorics of diagrams, via representation theory, or via the topology of the knot complement. We will touch on braid groups and mapping class groups, and use these groups to show that every (closed, orientable) 3-manifold can be constructed via knots. We will conclude by looking at knot complements via the tools of hyperbolic geometry.
Textbooks: We will draw material from the following sources. The selection of topics in Prasolov and Sossinsky is probably closest to the outline that we'll follow.
- An Introduction to Knot Theory, by W.B. Raymond Lickorish
- Knots, Links, Braids, and 3-Manifolds, by V.V. Prasolov and A.B. Sossinsky.
- Hyperbolic knot theory, by J. Purcell
Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.
Class Schedule and Homework
| Day | Topic | Reference | Homework |
|---|---|---|---|
| 8/27 | Definitions, Reidemeister moves | PS, §1 | |
| 8/29 | Tri-colorability and the fundamental group | L, p. 11, p. 110-112 | Homework 1, due 9/5 |
| 9/5 | Seifert surfaces | L, p. 15-18 | |
| 9/10 | The linking number | Rolfsen; Epple article | |
| 9/12 | Prime factorization | L, p. 19-21; Hedegard, p. 22-29 | Homework 2, due 9/19 |
| 9/17 | Alexander polynomial, part 1 | L, p. 49-51 | |
| 9/19 | Alexander polynomial, part 2 | L, p. 51-58 | |
| 9/24 | Skein relations, Kauffman bracket | PS, p. 23-28 | |
| 9/26 | Jones polynomial | PS, p. 29-32 | Homework 3, due 10/3 |
| 10/1 | Crossing number of alternating links | L, p. 41-45 | |
| 10/3 | Introduction to braids | PS, p. 47-52 | |
| 10/8 | Alexander and Markov theorems | PS, p. 54-60 | |
| 10/10 | Morton-Franks-Williams inequality | Article | Homework 4, due 10/17 |
| 10/15 | Braids and mappling class groups | PS, p. 61-65 | |
| 10/17 | Dehn-Lickorish theorem | PS, p. 90-93 | |
| 10/22 | Heegaard splittings of 3-manifolds | PS, p. 67-71, 75-77 | |
| 10/24 | Lens spaces, Dehn surgery | PS, p. 77-80, 84-86 | Homework 5, due 10/31 |
| 10/29 | Introduction to hyperbolic knots | P, Chapter 1 | |
| 10/31 | Hyperbolic structure on the figure-8 knot | P, Chapter 2 | |
| 11/5 | Hyperbolic structures on surfaces | P, Chapter 3 | |
| 11/7 | Developing map and completeness | P, Chapter 3 | Homework 6, due 11/16 |
| 11/12 | Gluing and completeness equations | P, Chapter 4 | |
| 11/14 | Gluing and completeness equations | P, Chapter 4 | |
| 11/26 | Completion and Dehn filling | P, Chapter 6 | |
| 11/28 | Presentation: Khanh, Rebekah | ||
| 12/3 | Presentation: Abeer, Dong Bin | ||
| 12/5 | Presentation: Rosie | ||
| 12/10 | Presentation: Kyle, Ben |
dfuter at temple edu Last modified: Fri Aug 21 13:41:22 PDT 2009