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.
・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
knot, twisted Alexander polynomial, colored Jones polynomial, matrix-weighted zeta function, volume of knot complement
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|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.|
References will be announced during the course.