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2020 Students Enrolled in or before 2015 School of Science Mathematics

Advanced courses in Geometry D

Academic unit or major
Mathematics
Instructor(s)
Tamas Kalman
Class Format
Lecture (Zoom)
Media-enhanced courses
-
Day of week/Period
(Classrooms)
3-4 Mon (H115)
Class
-
Course Code
ZUA.B334
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 [ZUA.B333 : Advanced courses 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, 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

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
  • ZUA.B301 : Geometry I
  • MTH.B341 : Topology
  • ZUA.B333 : Advanced courses in Geometry C

Prerequisites

Students are expected to have passed [Geometry I], [Geometry II], [Topology] and [Advanced courses in Geometry C].