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- Academic unit or major
- Mathematics
- Instructor(s)
- Kalman Tamas
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(H115)
- Group
- -
- Course number
- ZUA.B334
- Credits
- 1
- Academic year
- 2020
- Offered quarter
- 4Q
- Syllabus updated
- 2020/9/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
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 [ZUA.B333 : Advanced courses in Geometry C].
Student learning outcomes
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 that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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, Fox calculus | Details will be provided during each class session |
| Class 5 | Alexander polynomial II: Fox calculus, Conway skein relation, Kauffman states, equivalence of definitions | Details will be provided during each class session |
| Class 6 | Jones, Homfly, and two-variable Kauffman polynomials | Details will be provided during each class session |
| Class 7 | Morton's inequalities, Murakami--Ohtsuki--Yamada states | Details will be provided during each class session |
Out-of-Class Study Time (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
Assessment criteria and methods
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
- ZUA.B301 ： Geometry I
- MTH.B341 ： Topology
- ZUA.B333 ： Advanced courses in Geometry C
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have passed [Geometry I], [Geometry II], [Topology] and [Advanced courses in Geometry C].
