Wednesday 13:3015:30 Amado 919
This seminar will be an introduction to 3dimensional topology: knot theory and 3manifolds.

 I plan to start from the standard constructions in knot theory and slowly move towards more recent topics. Many topics are almost unrelated, so even if you miss or don’t understand one of them, you can still grasp the following ones. Depending on the time and the audience, I hope to cover at least some of the latest developments in knot theory (in particular, Khovanov and Knot Floer homology). Later, we will move on to 3manifolds, discussing various properties and constructions of 3manifolds (and, if time permits, some basic invariants of 3manifolds).
Prerequisites:

 I will try to keep the presentation always on an elementary level. There are no special prerequisites for this seminar, but a basic knowledge of linear algebra, groups given by generators and relations, and elementary topology would be definitely helpful.
Tentative syllabus:


 Knots and links; isotopy; diagrams; Reidemeister moves; colorings; writhe and linking numbers; prime decompositions

 Knot groups and quandles: algebra and geometry

 The Alexander polynomial: Fox calculus; Seifert surfaces; coverings

 Skein relations: AlexanderConway, Jones, and HOMFLY polynomials

 Other knotted objects: braids, string links, tangles, Legendrian knots

 Khovanov’s categorification of the Jones polynomial

 Vassiliev invariants; chord diagrams and relation to Lie algebras; Kontsevich integral

 3Manifolds; fundamental groups; prime decompositions

 Handle decompositions of 3manifolds and surgery

 Triangulations and Heegard decompositions

 Old and new invariants of 3manifolds

 HeegardFloer Homology

Grades:

 During (almost) every lecture I will formulate several problems. Final grades will depend on your activity in solving these problems. Also, few students may be asked to give talks on some subject.
Recommended Books:

 I will use different books for different topics. In particular, W.B.R. Lickorish,
Introduction to knot theory (Springer, 1997) will be extensively used.