Note from June 29, 2020
Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
5 credits
30.0 h + 15.0 h
Q2
Teacher(s)
Dos Santos Santana Forte Vaz Pedro; Lambrechts Pascal;
Language
English
Main themes
Knots and diagrams, the Jones and Alexander polynomial invariants, knots and the topology of surfaces, the fundamental group, special topics (knots and 3-lanifolds, homological invariants, applications of knot theoru to biology and chemistry), quantum invariants of knots and 3-manifolds, 2 and 4-manifolds, link homology and categorification.
Aims
At the end of this learning unit, the student is able to : | |
| 1 | Contribution of the course to learning outcomes in the Master in Mathematics programme. By the end of this activity, students will have made progress in:
The aim of the course is to present some advanced chapters in low-dimensional topology to prepare the student to a research carrer in a related domain. By the end of the course the student will be able to :
|
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Content
The course consists of an introduction of the basic langage and certain fundamental results of knot theory to study its interactions with other subjects, like the topology of 3-manifolds, physics and molecular biology. The following contents are to be addressed in the course.
- Basic notions. Definition of knot, knot projection and knot diagram, the Reidemeister moves, the 3-coloring, the linking number.
- The Jones polynomial. The kauffman bracket, the jones polynomial, the relation between the number of crossings of an alternating knot and the Jones polynomial.
-The topology of surfaces applied to knot theory. Seifert surfaces, the classification of surfaces with boundary, surgery on surfaces, the linking number as the intersection number with a Seifert surface.
-The knot genus. The additivity of the knot genus, prime knots and the unique decomposition of knots in terms of prime knots.
-The Alexander polynomial. Presentation matrices for modules, the Seifert matrix and the homology of the knot complement, the infinite cyclic cover of a link, the Alexander module, the Alexander polynomial and the genus.
-The fundamental group. The topology of the knot complement, the Wirtinger presentation, the p-coloring via the fundamental group.
-Special topics.
Examples :
- The 2-variable polynomials (HOMFLY-PT and Kauffman),
- The Jones polynomial as the Witten invariant
- Knots and 3-manifolds,
- 3- and 4-manifolds (Kirby calculus),
- Braids, tangles and the Temperley-Lieb algebra,
- Khovanov link homology,
- Applied knot theory (biology, chemistry, cryptography').
- Basic notions. Definition of knot, knot projection and knot diagram, the Reidemeister moves, the 3-coloring, the linking number.
- The Jones polynomial. The kauffman bracket, the jones polynomial, the relation between the number of crossings of an alternating knot and the Jones polynomial.
-The topology of surfaces applied to knot theory. Seifert surfaces, the classification of surfaces with boundary, surgery on surfaces, the linking number as the intersection number with a Seifert surface.
-The knot genus. The additivity of the knot genus, prime knots and the unique decomposition of knots in terms of prime knots.
-The Alexander polynomial. Presentation matrices for modules, the Seifert matrix and the homology of the knot complement, the infinite cyclic cover of a link, the Alexander module, the Alexander polynomial and the genus.
-The fundamental group. The topology of the knot complement, the Wirtinger presentation, the p-coloring via the fundamental group.
-Special topics.
Examples :
- The 2-variable polynomials (HOMFLY-PT and Kauffman),
- The Jones polynomial as the Witten invariant
- Knots and 3-manifolds,
- 3- and 4-manifolds (Kirby calculus),
- Braids, tangles and the Temperley-Lieb algebra,
- Khovanov link homology,
- Applied knot theory (biology, chemistry, cryptography').
Teaching methods
The main part of the course is given through lectures. During the sessions, students are asked to give suggestions and formulate ideas on the basis of their previous knowledge in order to further the course. Several examples of exercises will be given in order to illustrate and check understanding of subjects. The second part of the course consists of oral presentations by students on specially chosen topics.
Evaluation methods
Assessment is by a written test and by an oral presentation of one of the special topics, chosen by the student. These test knowledge and understanding of concepts, examples and fundamental results, ability to construct a coherent argument, and mastery of the techniques of proof introduced in the course. The examination has around five questions. Students may also choose the examination language (English or French).
Online resources
Page web du cours sur moodle
Bibliography
W. Lickorish: An Introduction to Knot Theory, GTM 175, Springer 1997.
J. Roberts: Knot knotes (disponible sur iCampus)
K. Murasugi: Knot theory and its applications, Birkhäuser, Springer 1996.
Teaching materials
- matériel sur moodle
Faculty or entity
MATH
