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.