Seminar in Topology 106384

Wednesday 13:30-15:30 Amado 919

This seminar will be an introduction to 3-dimensional topology: knot theory and 3-manifolds.

    • 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 3-manifolds, discussing various properties and constructions of 3-manifolds (and, if time permits, some basic invariants of 3-manifolds).


    • 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: Alexander-Conway, 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
      • 3-Manifolds; fundamental groups; prime decompositions
      • Handle decompositions of 3-manifolds and surgery
      • Triangulations and Heegard decompositions
      • Old and new invariants of 3-manifolds
      • Heegard-Floer Homology


    • 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.