Lecturer: José Pedro Quintanilha
Links: Müsli, heiCO
Lectures: Monday 09:15 - 10:45 in SR 3

This lecture is classified as a "Spezialisierungsmodul".

There are no exercise classes, but if you wish to participate in the course, please register on Müsli for the dummy tutorial.

Evaluation

Evaluation is through an oral examination at the end of the semester.

To take the exam, students need to register on heiCO and to contact me for scheduling the exam. Oral examinations will take place on Thursday, July 31 and Monday, August 4. Please indicate your choice among these dates and at what times you are available for the exam, at least one week in advance.

Additional examinations will happen in late August or September, at a date to be announced. If this is when you wish to take the exam, please also send me an email with that information, so I can contact you with the precise scheduling.

Program

Pre-requisites: algebraic topology (fundamental groups, coverings, singular (co)homology), basic differential topology (smooth manifolds, orientations).

  1. Introduction: knots, links, link isotopy, equivalent characterizations, link diagrams, Reidemeister's Theorem.
    Notes.
  2. Colorability of knots: p-colorings of a diagram, the Fox p-coloring space, invariance, computations, twist knots.
    Notes.
  3. Seifert Surfaces: oriented knots and links, Seifert's algorithm, the knot genus.
    Notes.
  4. The connected sum of knots: connected sum of manifold pairs, construction of the knot connected sum, properties of the connected sum, additivity of genus, prime knots, the prime decomposition theorem, the crossing number.
    Notes.
  5. Link exteriors: tubular maps and tubular neighborhoods, existence and uniqueness, well-definedness of the link exterior, first homology, meridians, Gordon-Luecke's Theorem.
    Notes.
  6. Linking phenomena: split links and their fundamental group, the linking number of knots, linking number via diagrams, symmetry, linking number as a Z-bilinear form on homology, Longitudes, Linking number from a Seifert surface.
    Notes.
  7. The fundamental group: the fundamental group and the unknot, the Loop Theorem, Wirtinger presentations, knot colorings via dihedral groups, non-split link exteriors as Eilenberg-MacLane spaces, the Sphere Theorem, the Generalized Schönflies Theorem, torus knots.
    Notes.
  8. The Alexander polynomial: the infinite cyclic covering, the Alexander module, the order ideal of a module, definition of the Alexander polynomial, Seifert matrices, properties of the Alexander polynomial, connection to the knot genus, the Alexander polynomial of torus knots.
    Notes.
  9. An introduction to Kirby calculus: handle decompositions of smooth manifolds, handle attachments via attaching spheres and framings, 1-handle attachments, 2-handle attachments to D⁴ as framed links, Kirby diagrams, representing closed oriented connected 4-manifolds, the linking matrix and the intersection form, handle moves.
    Notes.
    This is a bonus lecture, whose material will not be covered in the exam.

Literature

All material will be contained in the lecture notes, but here are some additional resources: