2020 Advanced topics in Geometry D

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kalman Tamas 
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
Mon3-4(Zoom)  
Group
-
Course number
MTH.B404
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 [MTH.B403 : Advanced topics 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 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

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
  • MTH.B331 : Geometry III
  • MTH.B403 : Advanced topics in Geometry C

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed [Advanced topics in Geometry C]

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