Topology, geometry and algebra of knots
       Fizmatclub, Autumn 2005

12 lectures

   1. Knots, links, braids, tangles.
Orientation. Knot diagrams. Reidemeister moves.
Invariant: number of arc colorings.
   2. Representation of a knot as a link closure.
Braid group: generators and relations, Artin's theorem.
Braid group on 3 strands as the group of the trefoil knot.
   3. Braids: Markov's theorem, Garside's theorem,
generators and relations for the group of pure braids.
Burau representation.
   4. Seifert surface. Genus of a knot.
Decomposition of a knot into primes.
Seifert form and Seifert matrix.
   5. Signature, determinant, Alexander polynomial,
Conway polynomial.
   6. Cobordism. Concordance. Slice knots. Alexander polynomial via
arc markings.
   7. The group of a knot. Wirtinger presentation.
Dehn's lemma. Gordon-Luecke theorem.
   8. Fox calculus. Relation between the group of a knot and Alexander's
polynomial. Quandle invariants. Presentations of modules.
   9. Coverings. Knot complement. Alexander duality.
Mayer-Vietoris theorem.
  10. Alexander covering. The group of a knot and knot inversion.
Theorem of Trotter about pretzel knots.
  11. The Jones polynomial. Tait's conjectures.
  12. Knot notation: Dowker's code, Conway's code.
Continued fractions and rational links.

Literature

C. Adams. The Knot Book.
Chmutov--Duzhin, CDBooK. Chapters 1, 2.
  http://www.pdmi.ras.ru/~duzhin/papers/cdbook0508.ps.gz
Duzhin--Chmutov, Uzly i ikh invarianty. Mat. prosveschenie, n. 3,
  1999, pp. 59-93 (in Russian). Online at
  http://www.pdmi.ras.ru/~duzhin/papers/onknots.ps.gz
Ch. Livingston. Knot theory.
Prasolov, Sossinsky. Knots, links, braids and 3-manifolds,
  Russian version available online  at www.mccme.ru/prasolov/.
R. Lickorish. An introduction to knot theory.
Joan Birman. Braids, links, and mapping class groups.
Crowell, Fox. Introduction to knot theory.
  (Best in Russian translation by A.M.Vinogradov -- because of ample
  translator's comments and because this edition also includes
  R.Fox's paper "A quick trip to knot theory").
K.Murasugi. Knot theory and its applications.
D.Rolfsen. Knots and Links.