Math 9024 Information

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.

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

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 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

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