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.