Knot theory seminar

Organisators Prof. Anna Beliakova, Prof. Viktor Schroeder and Mr. Jean-Marie Droz I can be contacted at jean-marie.droz@math.unizh.ch

The origins of knot theory

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

René Descartes view of Atoms

Lord Kelvin (26 June 1824 – 17 December 1907)

Tabulating knots Tait, Kirkman and Little

Gauss’ contribution The idea of linking number!

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

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…

The Jones polynomial Enabled mathematicians to solve long-standing conjectures

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

Organisation

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 e-mail, for ex. to fix another time of appointment)

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)

The subjects

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

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

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

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

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

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.

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

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

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

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

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

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

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

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