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Quantum groups and knot theory
(Spring 2010, vakcode: WI406046)
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.
Schedule: Thursday, 14:00-17:00, weeks 5-11, 13-18, 20.
Location:
Rec-B 2.40.
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.
Program:
Weekly we give here an update of the topics
treated during the class, and we list the homework.
Week 5: Definition of knots and links (topological,
smooth and PL), PL-structures on manifolds, equivalence of knots and
Schoenflies theorem.
Reference: Syllabus,
and Chapter 1 and 2 of Livingston's book
"Knot Theory" (see reference
[2] below).
Homework: no homework for next week.
Week 6: NEW: The
syllabus
of this week. Combinatorial equivalence of links,
Reidemeister's theorem, linking number of an oriented two component
link, mod p colorings of a link and algebraic definition of the
braid group.
Reference: Chapter 3 of Livingston's book "Knot Theory" (see reference
[2] below).
Homework-exercise: Compute the space of colorings mod p for the
trefoil knot and for the Hopf link (hand in: week 7, thursday february 18).
Week 7: Tensor product, Yang-Baxter equation, skein theory
and the Kauffman bracket.
Reference:
Syllabus.
Section 2 and 3 of Chapter 1, and 2.1-2.4 of
Chapter 5.
Homework-exercise: Compute the Kauffman backet of the
diagram of the figure eight knot (the second diagram in figure 1.1 of
Chapter 5).
Week 8: Topological braids, fundamental group and Hopf
algebras.
Reference: Section 1 of the
syllabus
and Chapter 2, subsections 1.1, 1.8, 1.9, 1.10 of the book.
Homework-exercise: Exercise (h) of the syllabus.
Week 9:
Section 2.1 of the book and 1.1-1.4, 1.7-1.9, 2,
Yoneda's lemma in relation
to universal properties and adjoint functors from the
syllabus
of this week.
Homework-exercise: Exercise k of the syllabus of week 8.
Week 10:
Paragraphs 1.5, 1.6 and 2.2 of syllabus week 9.
Paragraphs 1.1, 2.1-2.3 of
syllabus
of this week.
Homework-exercise: Exercise b of the syllabus of this week.
Week 11:
Syllabus
of this week treated, except Lemma 2.8.
Homework-exercise: Exercises 2.4 and 2.9.
Week 13: The
syllabus
of this week, treated up to Proposition 2.5.
Homework-exercise: Exercises 1.4, 1.9 and 2.3 of the syllabus
of this week.
Week 14: The
syllabus
of this week. Treated: the second half of the syllabus of last week
and the syllabus of this week up to Exercise 3.9.
Homework-exercise: Exercises 2.3 and 2.7 of the syllabus of this
week.
Week 15: The
syllabus
of this week.
Week 16: The
syllabus
of this week.
Homework: Read carefully the syllabus of week 16.
Week 17: The
syllabus
of this week. Treated: the material covered in the syllabi of weeks
15, 16 and 17. NEW: the syllabus of week 6 is now available,
see above!
Homework: Read carefully the syllabus of week 17.
Week 18: The
syllabus
of this week. Treated: sections 1 and 3 of the syllabus of this week.
Homework: Exercises 1.9 and 2.2 of the syllabus of this
week. Read carefully section 2 of the syllabus.
Week 19: no lectures.
Week 20: The
syllabus
of this week. The material treated in this last week
is section 2, 3 and parts of section 5 of the syllabus.
Homework: Read carefully section 5 of the syllabus of this week.
Exam: A grade will be determined based on
your solutions of a take home exam. The
take home exam is now available in pdf-format. You have to hand
in the solutions of the take home exam at latest on
monday, June 28, 2010. Please send the solutions to
us per email as pdf-file (if you have handwritten solutions, please
scan them and send the scans to us per email).
We also give homework on a regular basis.
It will be graded and will count positively for the final grade
(at most one extra point).
Additional literature:
[1] C. Kassel, Quantum Groups, Graduate
Texts in Mathematics 155, Springer Verlag.
[2] Charles Livingston, Knot Theory,
The Carus Mathematical Monographs, number 24.
[3] V.G. Turaev,
Quantum invariants of knots and 3-manifolds, W. de Gruyter, Berlin,
1994.
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