1. Knot theory seminar

2. Organisators
Prof. Anna Beliakova, Prof. Viktor Schroeder and Mr. Jean-Marie Droz
I can be contacted at

3. The origins of knot theory

4. Atomism
John Dalton (September 6, 1766 – July 27, 1844)

5. René Descartes view of Atoms

6. Lord Kelvin
(26 June 1824 – 17 December 1907)

7. Tabulating knots
Tait, Kirkman and Little

8. Gauss’ contribution
The idea of linking number!

9. Algebraic topology
Henri Poincaré April 29, 1854 – July 17, 1912

10. Classical knot theory
M.Dehn, J.W.Alexander, W.Bureau,O.Schreier, E.Artin, K.Reidemeister, E.R.Van Kampen, H.Seifert, J.H.C.Whitehead, H.Tietze, R.H.Fox…

11. The Jones polynomial
Enabled mathematicians to solve long-standing conjectures

12. 3-Manifold theory
During the sixties links became a way to study 3-manifolds, beginning a fruitful interaction between physic, geometry and knot theory.

13. Organisation

14. Each week, a (~90 minutes) talk will be given either by one person or two persons who, in that case, must also write a short hand out (which must be distributed before the end of the semester).
You can come in my office (Y36L82) Monday and Thursday between 15h and 17h to ask me questions and get advices concerning your presentation. It is advised to do so at least once. (You can also contact me by , for ex. to fix another time of appointment)

15. Material
« An introduction to knot theory » by W.B.R. Lickorish (abreviated [Li])
« Knots » by G.Burde and H.Zieschang (BZ)
« Knots and Links » by D.Rolfsen (Ro)

16. The subjects

17. First definitions
[BZ] Ch. 1 (not section c), elementary part of [Ro] Ch 1,2,4.
PL and Top categories
Def. of a a tame knot or link
Equivalence of equivalences theorem

18. Projecting knots [BZ] Ch.1 section c (perhaps Ch. 3) and [Li] Ch.1
Knot projections
Reidemeister Theorem
Comparison of knots in R^3 and S^3
A Proof of the fact that the trefoil is not
Trivial
(If time permits) The fundamental group of a knot

19. Seifert Surfaces and infinite covering
[Ro] Ch. 5,6,7 ,[BZ] Ch. 2,8, [Li] 2,7
(As time permits) Description of some simple knot invariants
Seifert surfaces
Short review of covering spaces
The infinite cyclic covering of a knot

20. The Alexander Polynomial
[Li] Ch. 6,7, [BZ] Ch. 8 and [Ro] Ch. 7,8
Definition of the Alexander polynomial via Infinite cyclic covering
Proof of its invariance

21. Knot factorisation [BZ] Ch. 2.C and Ch. 7 and [Li] Ch. 2
Sum and product of knots
Satellite knots
(As time permits) miscellaneous properties
of those constructions
Prime knots, factorisation of knots and
Properties of the factorisation

22. The Jones polynomial [Li] Ch. 3
Skein relations (in particular for the Alexander polynomial)
Def. Of the Jones polynomial
Invariance of the Jones pol.
Basic properties of the Jones pol.

23. Alternating links [Li] Ch. 4,5 Tait’s conjectures
Their proof via the Jones polynomial

24. 3-manifolds [Ro] Ch. 9 A,C,D,F,G and [LI] Ch. 12 Def. of 3-manifolds
Def. and examples of surgeries
Heegard splitting
Statement and explanation of the Kirby moves

25. Obtaining 3-manifolds through surgery
[Ro] Ch. 9.I and [Li] Ch. 12
Proof of the theorem of Lickorish and Wallace: Every closed connected oriented manifold can be obtained by surgery on S^3 along a link

26. A quantum invariant [Li] Ch.13 Def. of tangles Temperley-Lieb algebra
Def. of a 3-manifold from the Jones Polynomial

27. Calculations with quantum invariants
[Li] 14
Methods for calculating with quantum invariants
Calculation of the Jones polynomial of the torus knots

28. Braids
[BZ] Ch. 2.D Ch. 10
The many definitions of the braid group (by generators and relations, geometrical and as action on the free groups)
Braids can be used to represent links!
Solution of the word problem for braids

29. The HOMFLY-PT polynomial
[BZ] Ch.16 and [Li] Ch. 15,16
Def. and basic properties of the HOMFLY-PT polynomial, (through skein relation)
A little Hecke algebra and its connection to the HOMFLY-PT polynomial

30. The end
Thank you for your patience.
See you next week!