Special lectures on advanced topics in Mathematics I

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Hiroshi Goda / Tamas Kalman

Class Format

Lecture

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive

Class

-

Course Code

MTH.E533

Number of credits

200

Course offered

2021

Offered quarter

1Q

Syllabus updated

Jul 10, 2025

Language

Japanese

Syllabus

Course overview and goals

The Alexander polynomial is a basic tool in the study of knots. In this course we study the twisted Alexander polynomial, which is a generalization of the Alexander polynomial. We define the Alexander polynomial using the free differential of the fundamental group of the knot complement. It can be generalized by applying a linear representation of the fundamental group. I will explain that the twisted Alexander polynomial may be regarded as a matrix-weighted zeta function of a graph. The colored Jones polynomial also can be obtained from the graph, then I will introduce the recent topics on the volume of a knot complement.

We learn basic methods in knot theory. We understand the Alexander polynomial and the Jones polynomial, and then study some applications.

Course description and aims

・Be familiar with the fundamental group (knot group) of a knot complement
・Be familiar with the equivalence transformation of a knot group
・Be familiar with calculations of the Alexander polynomial and the Jones polynomial of a knot
・Construct an arc graph from a knot diagram, and be familiar with its matrix-weighted zeta function
・Understand the construction of the twisted Alexander polynomial and its applications

Keywords

knot, twisted Alexander polynomial, colored Jones polynomial, matrix-weighted zeta function, volume of knot complement

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Objectives

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.

References will be announced during the course.

Evaluation methods and criteria

Assignments (100%)

Related courses

• MTH.E645 ： Special lectures on current topics in Mathematics I
• ZUA.E343 ： Special courses on advanced topics in Mathematics I

Prerequisites

None required