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Quantum groups and knot theory

(Autumn 2011, vakcode: WI406046, Studielast: 6 ECTS)

The master course Quantum groups and knot theory is part of the master Mathematics and master Mathematical Physics at the University of Amsterdam.
Teachers: Eric M. Opdam and Jasper V. Stokman
Emails: e.m.opdam at uva.nl and j.v.stokman at uva.nl
Tel. 020-5255205 and 020-5255202
Course material:

C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, Panoramas et Syntheses, no. 5 (1997), Societe Mathematique de France, ISBN 2-85629-055-8

and additional texts, which will be downloadable from this website in due course. To order the book online, please go to the following website.

Lots of detailed information on particular knots and their invariants can be found in the Knot Atlas of Scott Morrison and Dror Bar-Natan. In particular have a look at
Schedule: Monday, 13:00-17:00, weeks 36-42, 44-50.
ANNOUNCEMENT: the lectures will remain on monday, following the above schedule!
Location: zaal G0.05 (weeks 36-42), NIKHEF, Science Park 105, zaal F2.19 (weeks 44-50).
Registration: In order to have access to the blackboard site and to be able to process your grade, we ask you to register for the course. If you are a UvA student, you can do so by sending an email (with your student number and the course information) to owb-rec-science AT uva.nl. If you are not registered as a UvA student, then you need to enroll first as guest student at studielink. You will receive a student number, with which you can register for the course as indicated above.
Exam: Unfortunately there is a short delay in making the exam available. The exam will now be made available on tuesday, january 17, 2012 before 12 pm, via the blackboard page of the course. If you are not able to enter the blackboard page, please contact us by email and we sent you the exam by email. This is an individual exam, it is not allowed to work together on the problems. You have to hand in the solutions before January 31, 2012, 10 am. If we are not in office you can put the solutions in one of our mailboxes. In that case, please send an email to us that you have done so. You can also send in your solutions by email, for instance by first scanning the solutions (make sure that the scans are readable though!), or by TeXing your solutions.

A final grade will be determined based on the grades for the homework exercises and for the take home examination as follows. At the end of the course a homework mark x between 0 and 1 will be determined, based on your solutions of the homework exercises. If the grade y of the take home examination is larger or equal to 5.5 then the final grade will be min(x+y,10). If y is less than 5.5 then you did not pass the exam and y will be the final grade. The precise date for the take home examination will be fixed shortly. The re-exam will be an oral examination with one of the teachers. In that case the homework will be discarded.
Program: Weekly we give here an update of the topics treated during the class and we list the homework. We will post also the lecture notes in due course. The blackboard page of this course is also available. Sometimes the information will be put on the blackboard page first and will be added to this homepage only at a later stage. So please always check the blackboard page if information seems to be missing on the homepage!

September 5 (week 36): The syllabus on Knots and Links, PL structures, PL-manifolds and Equivalence of Links is treated completely.
Homework: Read the syllabus.

September 12 (week 37): Here is the syllabus of this week. The whole syllabus has treated.
Homework: Exercises (a) and (c) of the syllabus of this week (the homework should be handed in at latest on monday, september 19, before the start of the lecture).

September 19 (week 38): The topics treated during the lectures are Yang-Baxter equations, skein theory and the Jones polynomial. Here is the corresponding syllabus. Please have a go at Exercise 3.5 in the syllabus at home. If you have questions about this exercise, please send an email to j.v.stokman at uva.nl.
Homework: The syllabus also contains the homework exercise of this week (the homework should be handed in at latest on monday, september 26, before the start of the lecture).

September 26 (week 39): Hopf algebras and Sweedler's notation. Here is the corresponding syllabus. Homework: Exercises (f)(i),(ii) and (iv) in the syllabus (the homework should be handed in at latest on monday, october 3, before the start of the lecture).

October 3 (week 40): Sections 1.1-1.6, 2.2 and 3 from the syllabus, Chapter 2, sections 2.1, 2.2, 2.5, 3.1-3.4 from [2], and braidings in non-strict monoidal tensor categories following [1], including the two hexagon axioms.
Homework:
(1) Exercises (d) and (e) from the syllabus (should be handed in at latest on monday, october 10 before the start of the lecture),
(2) Study example 2.3 and section 2.4 of [2] yourself.

October 10 (week 41): From syllabus week 40, Yoneda's lemma and representable functors. Also the syllabus of this week is discussed and Chapter 2 of [2].
Homework: Exercise (b) from the syllabus of this week (should be handed in at latest on monday, october 17 before the start of the lecture).

October 17 (week 42): The quantum double as a Hopf algebra, following the syllabus of this week.
Homework: Exercise 2.9 of the syllabus of this week (should be handed in at latest on monday, october 31 before the start of the lecture).

October 24 (week 43): No lectures (autumn break).

October 31 (week 44): The quantum double as a braided Hopf algebra, and the quantized universal enveloping algebra of sl(2), following the syllabus of this week (up to and including subsection 2.2).
Homework: Exercise 1.9 of the syllabus of this week (should be handed in at latest on monday, november 7 before the start of the lecture).

November 7 (week 45): The quantized universal enveloping algebra of sl(2) viewed as quotient of a quantum double (based on Subsection 2.3 of the syllabus of last week). We also discuss the associated R-matrix and braiding, following the syllabus of this week (up to and including Section 3). Section 4 of the syllabus will not be discussed in the class, and will not be part of the course material. It discusses the extension of the results to sl(n).
Homework: Exercises 2.3 and 2.7 of the syllabus of this week (should be handed in at latest on monday, november 14 before the start of the lecture).

November 14 (week 46): The syllabus of this week. Treated material: the syllabus, except for 3.4, 3.5, 4.2 and 4.4.
Homework: Read 3.4, 3.5, 4.2 and 4.4 (both in [2] and the additions in the syllabus, but you may skip the proof of Thm 3.17) by yourself,
and Exercise (i) of the syllabus (should be handed in at latest on monday, november 21 before the start of the lecture).

November 21 (week 47): The syllabus of this week.

November 28 (week 48): The syllabus of this week has been treated completely.
Homework: Read the syllabus yourself and Exercise (c) of the syllabus (should be handed in at latest on monday, december 5 before the start of the lecture).

December 5 (week 49): We discuss quantum invariants associated to sl(2). There are two syllabi this week, syllabus 1 and syllabus 2. The theory that we discuss matches with section 1 of the first syllabus and with sections 2 and 3 of the second syllabus. The rest of the two syllabi discusses techniques to compute quantum invariants, in particular techniques to compute the colored Jones polynomial using skein theory. This will not be treated in the course this year, but we advise you to read it through because it nicely combines the various methods treated in the course thusfar.
Homework: Exercise 1.9 of the first syllabus of this week (should be handed in at latest on monday, december 12 before the start of the lecture).

December 12 (week 50): The syllabus of this week. Sections 1 and 2 are treated.
Literature:
[1] C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer Verlag.
[2] C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, Panoramas et Syntheses, no. 5 (1997), Societe Mathematique de France.
[3] Charles Livingston, Knot Theory, The Carus Mathematical Monographs, number 24.
[4] V.G. Turaev, Quantum invariants of knots and 3-manifolds, W. de Gruyter, Berlin, 1994.