### 2021　Special courses on advanced topics in Mathematics I

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Mathematics
Instructor(s)
Goda Hiroshi  Kalman Tamas
Course component(s)
Lecture
Day/Period(Room No.)
Intensive ()
Group
-
Course number
ZUA.E343
Credits
2
2021
Offered quarter
1Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

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.

### Student learning outcomes

・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 that will be developed

 ✔ 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/Required learning

Course schedule Required learning
Class 1 The following topics will be covered in this order: -knot and knot group -Alexander polynomial -twisted Alexander polynomial -matrix-weighted zeta function of a graph -volume of a knot complement Details will be provided during each class session.

None required

### Reference books, course materials, etc.

References will be announced during the course.

### Assessment criteria and methods

Assignments (100%)

### Related courses

• MTH.E533 ： Special lectures on advanced topics in Mathematics I
• MTH.E645 ： Special lectures on current topics in Mathematics I

None required