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Quantum groups and knot theory 2013-2014

(6 EC)

The master course Quantum groups and knot theory is part of the master Mathematics 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
Rooms: C3.115 and C3.116, Science Park 904
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:

The lectures are on thursday mornings, weeks 36-42, 44-50. Unfortunately the lecture rooms will vary weekly, and three meetings will be at Roeterseiland in Amsterdam (REC-P rooms: Plantage Muidergracht 24. REC-JK rooms: Valckeniersstraat 65-67).
Note: the first and second week the lectures will start at 9:00, the other weeks at 10:00.

Thursday Sept. 5 (week 36): 9:00-11:00 in SP A1.10 and 11:00-12:00 in SP A1.08.
Thursday Sept. 12 (week 37): 9:00-12:00 in SP A1.06.
Thursday Sept. 19 (week 38): 10:00-13:00 in SP A1.14.
Thursday Sept. 26 (week 39): 10:00-13:00 in SP D1.162.
Thursday Oct. 3 (week 40): 10:00-13:00 in SP B0.208.
Thursday Oct. 10 (week 41): 10:00-13:00 in SP B0.209.
Thursday Oct. 17 (week 42): 10:00-11:00 in SP G3.02 and 11:00-13:00 in SP G3.13.
Thursday Oct. 31 (week 44): 10:00-13:00 in SP A1.10.
Thursday Nov. 7 (week 45): 10:00-13:00 in SP G3.13.
Thursday Nov. 14 (week 46): 10:00-11:00 in SP G3.02 and 11:00-13:00 in SP G2.13.
Thursday Nov. 21 (week 47): 10:00-13:00 in SP B0.201.
Thursday Nov. 28 (week 48): 10:00-13:00 in SP G3.02.
Thursday Dec. 5 (week 49): 10:00-11:00 in SP G2.10 and 11:00-13:00 in G2.02.
Thursday Dec. 12 (week 50): 10:00-13:00 in SP D1.116.
Thursday Dec. 19 (week 51): 10:00-13:00 in SP D1.162.
Registration:

It is mandatory to register for the course, see the course catalogue page for further details. If you are not registered as a UvA student, then you might need to enrol first as guest student at studielink.
Exam:

There will be weekly homework exercises, a final take-home exam and possibly a small oral exam about the solutions of the take-home exam you have handed in. Please note that the take-home exam is an individual exam. It is not allowed to work together on the exercises. Be aware that the exercises of the take-home exam will in general be harder than the homework exercises.

The take-home exam will be made available as pdf-file on the blackboard page of the course on Monday January 13 before 5 pm under Course Information. The solutions of the take-home exam should be handed in at latest Monday January 27 at 5 pm.
Possible ways to hand in: Electronically as a single pdf-file to both Jasper Stokman and Eric Opdam (please make sure that the size of the pdf-file does not exceed 5mb, lowering the resolution of the scans if necessary). A paper version of the solution set can be handed in directly to Jasper Stokman or Eric Opdam (in case we are not in our offices, please put it in one of our mailboxes). Please do not forget to write your name, student number and university on your solution set.

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 re-exam will be a written exam. 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 here in due course.

Note: All the information will be made available on this homepage, but 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!

September 5 (week 36): Paragraph 1 and 2, Chapter 1 of the book and from the syllabus up to and including Theorem 3.3.
Homework: Exercise 1.5a of the book (the homework should be handed in at latest on thursday, september 12, before the start of the lecture).

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

September 19 (week 38): from the book Chapter 2: subsections 1.1, 1.3, 1.4, 1.8, 1.9, 1.10 and subsections 2.1, 2.3, 2.5. Here is the syllabus of this week.
Recommended: read the above parts of the book and look at exercises (f), (g), (h) and (i) of the syllabus.
Homework: Exercise (k) of the syllabus.

September 26 (week 39): Section 1 and section 2 from the syllabus of this week.
Recommended: carefully read sections 1 and 2 of the syllabus and Chapter 2, sections 1 and 2 of the book.
Homework: 1. Prove formula (1.6) from Chapter 2 of the book yourself by induction (coproduct of T(V)). Show furthermore that the algebra homomorphism S from T(V) to T(V)^{op}, characterised by S(v)=-v for v in V, defines an antipode.
2. Exercise 1.7(b) of Chapter 2 of the book.

October 3 (week 40): The syllabus of this week is treated.
Recommended: study carefully the proof of Proposition 2.7 of the syllabus.
Homework: From Chapter 2, Section 4.4 of the book: Exercise (b).

October 10 (week 41): The syllabus of this week is treated.
Homework: Exercise 2.9 in the syllabus of this week.

October 17 (week 42): The syllabus of this week is treated up to and including subsection 2.2.
Homework: Exercise 1.9 in the syllabus of this week.

October 24 (week 43): Autumn break.

October 31 (week 44): The remaining part of the syllabus of week 42 is treated, as well as sections 1-3 of the new syllabus.
Homework: Study carefully sections 1-3 of the syllabus of this week.
Hand-in homework: Exercises 2.3 and 2.7 of the syllabus of this week.

November 7 (week 45): The syllabus of this week. Treated material: sections 1,2,3.1-3.3,4.1,4.2.
Homework: Read carefully section 4.1 and the proof of Theorem 4.22 of the syllabus.
Hand-in homework: Exercise (g) and (i) of the syllabus.

November 14 (week 46): The syllabus of this week. Treated material: Syllabus week 45, Section 3 and from Section 4 Thm. 4.22, 4.23 and 4.24. Syllabus week 46, Section 1 and Subsection 2.1.
Homework: Read carefully subsection 2.1 of the syllabus of this week.
Hand-in homework: Exercise (a) and (b) of the syllabus of this week. Do it algebraically and give also the derivation using diagrams.

November 21 (week 47): The syllabus of this week. Treated material: the rest of the syllabus of week 46, and from the syllabus of week 47 up to and including Definition 1.6.
Homework: Exercise (i) of the syllabus of week 46.

November 28 (week 48): The syllabus of this week. Treated material: the rest of the syllabus of week 47, and from the syllabus of week 48 up to and including Lemma 3.5.
Homework: read yourself in the syllabus of week 48 up to and including Theorem 3.10 and its proof.
Hand-in homework: Exercise c of the syllabus of week 47.

December 5 (week 49): Section 1 and 2 of the syllabus of last week. On December 11 a new version of the syllabus of week 48 is put on the website, with some small corrections.

December 12 (week 50): We discuss quantum invariants associated to sl(2). There are two syllabi this week, syllabus 1 and syllabus 2. The theory that we discuss corresponds with section 1 of the first syllabus and sections 2 and 3 of the second syllabus.

December 19 (week 51): Two extra topics are discussed:
(1) Computing the coloured Jones polynomial (second syllabus of last week, section 5).
(2) Quantum groups and the Heisenberg XXZ spin chain.

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.