Дистанционный курс Владимира Вершинина "Braids and Knots"

Braids and Knots

TIME: Mondays 5pm – 8pm
FIRST CLASS: April 6

HOW TO ENROLL: please write to the instructor, Vladimir Verchinine vladimir.verchinine–at--umontpellier–dot--fr. If you would like to earn ECTS credits for this course, please inform Svetlana Balaeva sbalaeva–at--hse–dot--ru – it is not guaranteed that the course can be made creditable but it may be considered.
 
PREREQUISITS: Basics of Algebra, Linear Algebra and Analysis (university first year)
 
SYLLABUS:
1. Definition and general properties of knots and  links.
2. Piecewise-linear maps. Polyhedra. PL-manifolds.
3. Definitions of braids, tangles and configuration spaces.
4. Regular knot projections. Diagrams of knots.
5. Reidemeister moves. Linking number. Seifert surface, genus of a knot.
6. Presentation of the braid group, the group of pure braids, Markov normal form.
7. Automorphisms of a free group and mapping class groups.
8. Algebraic properties of the braid group, Garside's theorems.
9. Representation of braids, Bigelow-Krammer theorem.
10. Dehornoy's theorem on the ordering of braids.
11. Wirtinger presentation the knot group.
12. Alexander's theorem (Vogel's algorithm).
13. Markov's theorem.
14. Conway and Jones polynomials for links.
15. Vassiliev invariants of knots.
16. Kauffman brackets and Kauffman's formula for Jones polynomial.
17. Khovanov homology.