Quantum groups and knot theory 20132014
(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. 0205255205 and 0205255202
Course material:
C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants,
Panoramas et Syntheses, no. 5 (1997), Societe Mathematique de France,
ISBN 2856290558
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 BarNatan.
In particular have a look at

the
manual of the Mathematica Package KnotTheory,

the
Rolfsen Knot Table of 165 mathematical knots with at most 10 crossings,
including detailed (mathematical) information about each knot.
Schedule:
The lectures are on thursday mornings, weeks 3642, 4450. Unfortunately the lecture rooms will vary weekly, and three meetings will be at Roeterseiland in Amsterdam (RECP rooms: Plantage Muidergracht 24. RECJK rooms:
Valckeniersstraat 6567).
Note: the first and second week the lectures will start at 9:00, the other weeks at 10:00.
Thursday Sept. 5 (week 36): 9:0011:00 in SP A1.10 and 11:0012:00 in SP A1.08.
Thursday Sept. 12 (week 37): 9:0012:00 in SP A1.06.
Thursday Sept. 19 (week 38): 10:0013:00 in SP A1.14.
Thursday Sept. 26 (week 39): 10:0013:00 in SP D1.162.
Thursday Oct. 3 (week 40): 10:0013:00 in SP B0.208.
Thursday Oct. 10 (week 41): 10:0013:00 in SP B0.209.
Thursday Oct. 17 (week 42): 10:0011:00 in SP G3.02 and
11:0013:00 in SP G3.13.
Thursday Oct. 31 (week 44): 10:0013:00 in SP A1.10.
Thursday Nov. 7 (week 45): 10:0013:00 in SP G3.13.
Thursday Nov. 14 (week 46): 10:0011:00 in SP G3.02 and
11:0013:00 in SP G2.13.
Thursday Nov. 21 (week 47): 10:0013:00 in SP B0.201.
Thursday Nov. 28 (week 48): 10:0013:00 in SP G3.02.
Thursday Dec. 5 (week 49): 10:0011:00 in SP G2.10 and
11:0013:00 in G2.02.
Thursday Dec. 12 (week 50): 10:0013:00 in SP D1.116.
Thursday Dec. 19 (week 51): 10:0013:00 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 takehome exam and possibly a small
oral exam about the solutions of the takehome exam you have handed in. Please note that the takehome exam is an individual exam. It is not
allowed to work together on the exercises. Be aware that the exercises of the takehome exam will in general be harder than the homework exercises.
The takehome exam will be made available as pdffile on the
blackboard page of the course
on Monday January 13 before 5 pm under Course
Information. The solutions
of the takehome exam should be handed in at latest
Monday January 27 at 5 pm.
Possible ways to hand in: Electronically as a single pdffile to both
Jasper Stokman and Eric Opdam (please make sure that the size of the pdffile 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 takehome 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 reexam 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 13 of the new
syllabus.
Homework: Study carefully sections 13 of the syllabus of this week.
Handin 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 1,2,3.13.3,4.1,4.2.
Homework: Read carefully section 4.1 and the proof of Theorem 4.22
of the syllabus.
Handin 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.
Handin 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.
Handin 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 3manifolds, W. de Gruyter, Berlin,
1994.