Dror Bar-Natan: Classes: 2001-02:

Seminar on Knot Theory

Instructor: Dror Bar-Natan, drorbn@math.huji.ac.il, 02-658-4187.

Meetings: Mondays 12:00-14:00 at Upper Papik.

Office hours: Tuesdays 14:00-15:00 in my office, Mathematics 309.

Agenda: Have every student give at least one fun lecture on elementary knot theory.

Prerequisites: Meant for advanced undergraduate students.

Reading material:

  • W.B. Raymond Lickorish, An Introduction to Knot Theory, GTM 175, Springer-Verlag, New York 1997.
  • The books by Kauffman and Rolfsen.
  • V. V. Prasolov and A. B. Sossinsky's Knots, links, braids and 3-manifolds: an introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, American Mathematical Society 1997.

Thanks, Raz, for the tushim!

Some suggested topics for student lectures: (more may be added later, and you may choose subjects not on this list!)

Topic Speaker and Date Details Dependencies
Reidemeister's theorem Lior Zaibel, March 18th Prove that any two diagrams for the same knot are connected by a sequence of Reidemeister moves. See almost any book on knot theory. None.
Linking and self-linking numbers   The simplest measurement of the linking of a two component link and its problemmatic nature in the case of a single component knot. See almost any book on knot theory. None.
Seifert surfaces and knot factorization Ittai Chorev, April 29th Knots are like integers - there is a "product", there are "primes", and there is "unique factorization". See Lickorish's book. None.
The Kauffman bracket and the Jones polynomial Yossi Regev, April 15th These are very simple and very powerful knot invariants. See almost any modern book on knot theory. None.
Khovanov's categorification of the Jones polynomial   A new and very surprising extension of the Jones polynomial using homological algebra. Hot but difficult! See my paper On Khovanov's Categorification of the Jones Polynomial. The Jones polynomial
The geometry of alternating links Tom Erez, June 10 These are links whose diagrams alternate up-down-up-down... They are particularly well behaved. See Lickorish's book. Seifert surfaces and factorization
The Jones polynomial of alternating links Tomer Avidor, May 6 See Lickorish's book. The Jones polynomial
The Alexander polynomial   A classical invariant deeply rooted in algebraic topology. See Lickorish's book and many other sources. Seifert surfaces, linking numbers.
The fundamental group   The fundamental group of a knot complement is an extremely strong invariant of the knot, it's easy to compute, but... See any book on knot theory. None.
Surgery and 3-manifolds   This is one of the fundamental links between knot theory and "higher" 3-dimensional topology; every 3-manifold comes from a knot in some way! See Lickorish's book and many other sources. None.
Finite type invariants and weight systems   These are knot invariants that behave as if they are polynomials on the the space of all knots. They seem very strong, but nobody really knows how strong they are. See my paper On the Vassiliev knot invariants and many further sources in VasBib. None, but the Jones or Alexander polynomials are helpful.
Weight systems and Lie algebras   One reason why finite type invariants are so interesting is that they are intimately connected to Lie algebras (via their weight systems). See my paper On the Vassiliev knot invariants and many further sources in VasBib. Finite type invariants
The second hull of a knotted curve Aviv Sheyn, May 27 Every knot defines a "deep inside" part of space which is "surrounded" by the knot more than just once. See SecondHull.gif and the article The Second Hull of a Knotted Curve by Cantarella, Kuperberg, Kusner and Sullivan. None
Braids and knots Yishai Fried, June 24 The theorem stating that every knot is the closure of a braid and Markov's theorem, a complete description of knots in terms of braids. See the Prasolov-Sossinsky book. None

Weekly schedule:

March 11 Dror Bar-Natan: A quick introduction to knots and knot invariants. Handouts: Classes: 2001-02: Knots and Feynman Diagrams: Some Non Obvious Examples and Classes: 2000-01: Knot Theory: Pathologies. See also http://home.pacbell.net/bullnose/rebarfaq.htm.
March 18
Lior Zaibel on Reidemeister's theorem. Handouts: Zaibel-1.gif, Zaibel-2.gif, Zaibel-3.gif.
March 25 Passover vacation
April 1 Passover vacation
April 8 Class cancelled - Dror was sick.
April 15
Yossi Regev on The Kauffman bracket and the Jones polynomial. Handouts: Regev-1.gif, Regev-2.gif, Regev-3.gif.
April 22 Student lectures: Lior Zaibel continued his lecture and Yossi Regev finished his.
April 29
Ittai Chorev on Seifert surfaces and knot factorization. Handouts: Chorev-1.gif, Chorev-2.gif, Chorev-3.gif, Chorev-4.gif.
May 6
Tomer Avidor on the Jones polynomial of alternating links. Handouts: Avidor-1.gif, Avidor-2.gif.
May 13 Student lecture: Ittai Chorev continued his lecture.
May 20 Student lecture: Tomer Avidor continued his lecture.
May 27
Aviv Sheyn on the second hull of a knotted curve.
June 3 Aviv Sheyn will continue his lecture and I will talk about finite type invariants a little.
June 10
Tom Erez on the geometry of alternating links. Exhibits: Erez-1.jpg; Handouts: Erez-2.png, Erez-3.png, Erez-4.png, Erez-5.png.
June 17 Student lecture: Tom Erez continued his lecture.
June 24
Yishai Fried on braids and knots.

Background image: A table of knots and links by Rob Scharein, taken from http://www.cs.ubc.ca/nest/imager/contributions/scharein/zoo/knotzoo.html.