2020 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry D
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tamas Kalman
- Class Format
- Lecture (Zoom)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (Zoom)
- Class
- -
- Course Code
- MTH.B404
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
The aim of this lecture course is to familiarize students with the basic language of and some fundamental theorems in knot theory.
This course is a continuation of [MTH.B403 : Advanced topics in Geometry C].
Course description and aims
Students are expected to
・be able to show the equivalence of some knots and, via the use of invariants, the inequivalence of others
・understand the construction of some of the most commonly used knot polynomials.
Keywords
knot, link, knot group, genus, Alexander, Jones, and Homfly polynomials, infinite cyclic cover, Seifert matrix
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | definition and examples of knots and links, diagrams, Reidemeister moves | Details will be provided during each class session |
| Class 2 | knot group, Wirtinger presentation, Seifert surface, genus | Details will be provided during each class session |
| Class 3 | connected sum, prime decomposition | Details will be provided during each class session |
| Class 4 | Alexander polynomial I: infinite cyclic cover, Seifert matrix | Details will be provided during each class session |
| Class 5 | Alexander polynomial II: Fox calculus, Conway skein relation, Kauffman states | Details will be provided during each class session |
| Class 6 | Alexander polynomial III: equivalence of definitions | Details will be provided during each class session |
| Class 7 | Jones, Homfly, and two-variable Kauffman polynomials | Details will be provided during each class session |
| Class 8 | Morton's inequalities, Murakami--Ohtsuki--Yamada states | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
C. Livingston: Knot Theory
D Rolfsen: Knots and links
Evaluation methods and criteria
Evaluation will be based on exams and homework. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B403 ： Advanced topics in Geometry C
Prerequisites
Students are expected to have passed [Advanced topics in Geometry C]