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
Textbooks: We will draw material from the following sources, in addition to miscellaneous articles. Prerequisites: Math 9023 or permission of the instructor.

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.

Class Schedule and Homework

This table will be gradually filled in as the course progresses. The letters H, L, P, and T stand for the above references. FM stands for Farb-Margalit, CB for Casson-Bleiler.

Day Topic Reference Homework
1/13 Polyhedral decomposition for alternating knotsP, 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 SFSOrbifolds; 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
3/31Nielsen-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
4/14Stable & unstable foliations CB, P. 89-94
4/16Transverse measures CB, P. 95-102
4/21Presentation: Thomas
4/23Presentation: Zach
4/28Presentation: Will, Tim
4/30Presentation: Geoff

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Last modified: Fri Aug 21 13:41:22 PDT 2009