Math 9024: Knot Theory and Low-Dimensional Topology
Fall 2014
| Meets: | Tue/Thu 9:30 AM - 10:50 AM in Wachman Hall, room 1015D |
| Instructor: | David Futer |
| Office: | 1038 Wachman Hall |
| Office Hours: | by appointment |
| E-mail: | dfuter at temple.edu |
| Phone: | (215) 204-7854 |
Course content: This course will be a continuation of Math 9023. We will focus somewhat more on the geometric side of knot theory and 3-manifold theory. Some topics include:
- Geometric topology of alternating knots
- Fibrations of 3-manifolds over the circle
- Nielsen-Thurston classification of mapping classes
- Geometric structures on 3-manifolds
- Hyperbolic geometry
- Volume conjecture
- Automorphisms of Surfaces after Nielsen and Thurston, by Andrew Casson and Steven Bleiler
- Notes on Basic 3-Manifold Topology, by Allen Hatcher
- An Introduction to Knot Theory, by W.B. Raymond Lickorish
- Hyperbolic Knot Theory, by Jessica S. Purcell
- Three-Dimensional Geometry and Topology, by William P. Thurston
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 |
|---|---|---|---|
| 1/13 | Polyhedral decomposition for alternating knots | P, p. 9-14 | |
| 1/15 | Normal surfaces, incompressible surfaces | L, p. 32-36 | |
| 1/20 | Surfaces in alternating knot complements | L, p. 36-38 | |
| 1/22 | Basic hyperbolic geometry | P, p. 19-21 | Homework 1, due 1/29 |
| 1/29 | Hyperbolic geometry | T, p. 53-64 | Homework 2, due 2/5 |
| 2/3 | Hyperbolic surfaces | T, p. 47-48, 86-90. | |
| 2/5 | Completeness of surfaces | P, p. 34-39. T, p. 147-150. | Homework 3, due 2/12 |
| 2/10 | Hyperbolic 3-manifolds, (G,X) structures | P, p. 27-31. T, p. 110-115 | |
| 2/12 | Developing map, holonomy, completeness | P, p. 32-34, 39. T, p. 139-146. | |
| 2/17 | Gluing equations for 3-manifolds | P, p. 45-48. | |
| 2/19 | Gluing and completeness equations | P, p. 48-52. | Homework 4, due 2/26 |
| 2/24 | Mostow rigidity | Benedetti-Petronio | |
| 2/26 | Hyperbolic Dehn surgery | P, p. 56-61 | Homework 5, do over the break |
| 3/10 | Model geometries | T, p. 179-189 | |
| 3/12 | Seifert fibrations, orbifolds | Wikipedia on SFS | Homework 6, due 3/19 |
| 3/17 | Geometric orbifolds and SFS | Orbifolds; Selberg's lemma | |
| 3/19 | Sphere and torus decomposition | H, p. 6-16 | |
| 3/24 | Geometrization theorem | Wikipedia; FM, p. 400-401. | |
| 3/26 | Measured foliations, pseudo-Anosovs | FM, p. 314-320. | Homework 7, due 4/2 |
| 3/31 | Nielsen-Thurston theorem; criteria for pseudo-Anosovs | FM, p. 397-399; 420-423 | |
| 4/2 | Geodesic laminations | CB P. 60-69 | |
| 4/7 | Constructing the stable lamination | CB, p. 79-83 | |
| 4/9 | Unique stable & unstable laminations | CB, p. 83-87 | |
| 4/14 | Stable & unstable foliations | CB, P. 89-94 | |
| 4/16 | Transverse measures | CB, P. 95-102 | |
| 4/21 | Presentation: Thomas | ||
| 4/23 | Presentation: Zach | ||
| 4/28 | Presentation: Will, Tim | ||
| 4/30 | Presentation: Geoff |
dfuter at temple edu Last modified: Fri Aug 21 13:41:22 PDT 2009