A knot is a knotted ring in three-dimensional space. Knot theory is a research field that targets knots. In this lecture,
you will learn basic notions of knot theory and its application in topology. In particular, deepen the understanding of the
knot table and various invariants, which are indispensable when considering knots. Knot theory is used not only in the natural
sciences but also in applied research. Research is being conducted on new anticancer drugs and antibiotics that prevent the
knot from being untied. It is useful for the students of this major to acquire such a concept. The main contents to be dealt
with are as follows.
- To be able to understand Reidemeister moves.
- To be able to understand notation for knots
- Understand classical invariants and be able to solve exercises related to them.
- Understand polynomial invariants and be able to solve exercises related to them.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1． | Basics on knots and links | Check this syllabus. | 10minutes |
| Confirm the definitions of knots and links. | 70minutes | ||
| Solve problems that I asked in class. | 100minutes | ||
| 2． | Reidemeister moves | Confirm the definitions of Reidemeister moves. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| Understand knots and links visually. | 70minutes | ||
| 3． | Unknotting operations | Confirm the definition of unknotting operations. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| 4． | Colorability and coloring number 1 | Confirm the definition of colorability and coloring number. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| 5． | Colorability and coloring number 2 | Learn how to calculate the coloring number. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| Consider the proof of the theorem on the coloring number. | 60minutes | ||
| 6． | Knot table and the Dowker notation for knots | Examine the knot table and see some concrete examples. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| Confirm the definition of the Dowker notation for knots | 70minutes | ||
| 7． | Conway's notation, knots and planar graphs | Confirm the difference between the Dowker notation and the Conway's notation. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| Understand the relationship between knots and planar graphs. | 70minutes | ||
| 8． | Classical invariants (unknotting number, bridge index, intersection number, etc.) | Confirm the definition of invariants for knots. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| Being able to calculate the unknotting number, bridge index, intersection number, etc. | 70minutes | ||
| 9． | Seifert surfaces | Confirm the definition of Seifert surfaces. | 20minutes |
| Solve problems that I asked in class. | 170minutes | ||
| Being able to draw Seifert surfaces for some knots. | 70minutes | ||
| 10． | Standard forms of Seifert surfaces | Solve problems that I asked in class. | 190minutes |
| Confirm the definition of standard forms of Seifert surfaces. | 40minutes | ||
| Review how to draw standard forms of Seifert surfaces. | 60minutes | ||
| 11． | Seifert matrix | Confirm the definition of Seifert matrixes. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| 12． | Conway polynomial 1 | Confirm the definition of the Conway polynomial. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| 13． | Conway polynomial 2 | Confirm the properties of the Conway polynomial. | 90minutes |
| Solve problems that I asked in class. | 100minutes | ||
| 14． | Polynomial invariants (Jones polynomials, HOMFLY polynomials, Kauffman polynomials, etc.) | Confirm the definitions of some polynomial invariants. | 190minutes |
| Investigate the properties of polynomial invariants. | 40minutes | ||
| Solve problems that I asked in class. | 60minutes | ||
| Total. | - | - | 3260minutes |
| レポート | Total. | |
|---|---|---|
| 1. | 25% | 25% |
| 2. | 25% | 25% |
| 3. | 25% | 25% |
| 4. | 25% | 25% |
| Total. | 100% | - |
Conduct reports and exercises several times, and pass 60% or more of the total score. The acceptance criteria (60%) are that
"selection of the method to be applied" and "calculation to reach the solution" can be executed almost correctly for a given
problem.
No textbook is specified, but the lessons will be conducted based on the board and handouts.
References: Knot Theory and Its Applications, by Kunio Murasugi
References: Knot Theory and Its Applications, by Kunio Murasugi
Basic knowledge of knot theory is not a prerequisite, but it is desirable to be able to use proof methods such as mathematical
induction and reductio ad absurdum.
- We accept questions and consultations during office hours.
- Course that cultivates an ability for utilizing knowledge
- Course that cultivates a basic interpersonal skills
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | Not applicable |
Last modified : Wed May 12 04:46:31 JST 2021