Math 779

Introduction to Knot Theory and 3-Dimensional Topology

Spring Semester, 2005

Course Information:


This course will cover basic results on 3-manifolds and knot theory. In the first part of the semester, I will talk about basic properties of knots, and I will introduce some knot invariants (tools to check if two knots are "different"). In the second part of the semester, I will go over the prime and the torus decompostions of 3-manifolds, the classification of Seifert fibered spaces, and hopefully some normal surface theory. There are no prerequisites for this class, even though a background in algebraic topology (fundamental group and homology) will make your life easier at times. All you need to have for this class is a good imagination, since most of 3-manifolds don't live in R^3 but they are made of pieces in R^3 glued together.
I plan to structure the class as follows: two classes a week of 75 minutes each of which will be composed of a 50 minute lecture and a 25 minute problem session in groups of 4 or 5.
For this class, you will be expected to turn in:
  • a set of problems due every 2 or 3 weeks (which will include the problems in the problem sessions)

    • or
  • either a presentation of an article (or part of an article) or a 4-5 page paper about an open problem in 3-manifold or knot theory (you don't have to figure out an open problem!).



    • I have listed below a few articles in mind which are fairly easy to read. If you want to present your own article you will need my approval. Here's a list of proposed articles:
  • Algorithms for recognizing knots and 3-manifolds, Joel Hass
  • Decision problems in the space of Dehn fillings, William Jaco and Eric Sedgwick
  • An algorithm to recognize the 3-sphere, Hyam Rubinstein
  • Connected sums of closed orientable triangulated 3-manifolds, Alexander Barchechat
  • The computational complexity of knots and link problems, Joel Hass, Jeffrey Lagarias, and Nicholas Pippenger
  • Homeomorphisms of non-orientable two-manifolds, W.B.R. Lickorish
  • The geometries of 3-manifolds, Peter Scott
  • Representations of 3-manifold groups, Peter Shalen



    • If you are interested in turning in a 4-5 page paper on an open problem, go to http://math.berkeley.edu/~kirby/ and click on "Problems in Low Dimensional Topology".

    Suggested readings:

    "The knot Book", Colin Adams
    "Knots and links", Dale Rolfsen