授業の目的 【日本語】
Goals of the Course(JPN) | | |
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授業の目的 【英語】
Goals of the Course | | | This course introduces fundamental concepts of knot theory, focusing on the classification of knots and links in three-dimensional space using two-dimensional diagrammatic methods. It extends these ideas to knots and links in the thickened torus and periodic tangles, examining how equivalence relations and invariants are constructed in these settings. Students will explore numerical invariants such as crossing number, unknotting number, and linking number, along with the development of the Jones polynomial. The course also highlights applications in various fields, including physical systems, computer science (AI and machine learning), textiles, medical science, and molecular chemistry. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | |
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到達目標 【英語】
Objectives of the Course | | By the end of this course, students will be able to:
1. Understand fundamental concepts of knots and links, their equivalence relations, and classification methods.
2. Apply two-dimensional diagrammatic theory to translate three-dimensional knot problems into combinatorial problems.
3. Compute numerical invariants (crossing number, unknotting number, linking number) and recognize their limitations in classification.
4. Understand the construction of the Jones polynomial and apply it to links in three-space and the thickened torus.
5. Extend the classical theory to doubly periodic tangles and explore its interdisciplinary applications. |
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授業の内容や構成
Course Content / Plan | | Title: Knot Theory, Periodic Tangles & Interdisciplinary Applications
(結び目理論、周期的タングルと学際的応用)
1. General Introduction: Definition of knots and links, classification problem, planar projections and knot tables, extension to three-manifolds, periodic tangles, interdisciplinary applications.
2. Diagrammatic Theory: The classical Reidemeister theorem, extension to links in the thickened torus, doubly periodic tangles, and real-world applications.
3. Basic Numerical Invariants for Classification: Crossing number, unknotting number, linking number, limitations, and interdisciplinary applications.
4. Introduction to the Jones Polynomial: Historical background, from the bracket polynomial to the Jones polynomial, invariance, and applications.
5. Extension of the Jones Polynomial to doubly periodic tangles: periodic boundary conditions, invariance, and applications.
6. Summary and Advanced Discussions: Review of the key points of the course, introduction to other invariants, extensions to singly and triply periodic structures, case studies (polymers, chemistry, textiles, etc.). |
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履修条件
Course Prerequisites | | There are no strict prerequisites for this course. However, a basic understanding of topology and linear algebra can be useful.
この講義は英語で行います。
This course will be taught in English. |
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関連する科目
Related Courses | | | No strict recommendation. |
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「履修取り下げ届」提出の要・不要
Necessity / Unnecessity to submit "Course Withdrawal Request Form" | | | Students do not need to submit a formal withdrawal request (履修取り下げ届は不要). |
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履修取り下げの条件等
Conditions for Course Withdrawal | | | Students who wish to withdraw from the course should consult the instructor in advance. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | 1. Class Participation & Discussions (30%) – Attendence and engagement in discussions.
2. Final Report (70%) – Students will solve problems and/or write a report applying course concepts to a given topic. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | W (Withdrawal): Given if a student withdraws before the final report deadline.
F (Failure): Given if a student fails to submit the final report or does not meet the passing criteria. |
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参考書
Reference Book | | - C. Adams, The knot book: an elementary introduction to the mathematical theory of knots, (American Mathematical Society, Providence, 2004) （結び目の数学 / 金信 泰造 訳）
- A. Kawauchi, A survey of knot theory, (Birkhauser, Basel 1996)
- K. Murasugi, Knot theory and its applications, (Birkhauser, Boston 2008). |
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教科書・テキスト
Textbook | | |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Students are encouraged to review lecture materials and explore suggested exercises to deepen their understanding. While homework is not required, optional exercises will be provided for independent study. Questions can be addressed outside class hours. A final report is required, where students will apply course concepts to some selected problems. |
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注意事項
Notice for Students | | | This course is designed for both 4th year undergraduate and graduate students. The level of discussion may vary depending on the audience, and students are encouraged to ask questions to clarify concepts. |
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他学科聴講の可否
Propriety of Other department student's attendance | | | Students from other departments are welcome to enroll in this course, provided they have an interest in the subject. No prior knowledge of knot theory is required, but a basic understanding of topology and algebra will be beneficial. |
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他学科聴講の条件
Conditions for Other department student's attendance | | | There are no strict restrictions for students from other departments. However, familiarity with basic topology and algebra is useful to follow the course effectively. Interested students are welcome and should ensure they are comfortable with mathematical reasoning and are encouraged to ask questions to clarify concepts. |
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レベル
Level | | |
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キーワード
Keyword | | | Knot theory, knots, links, invariants, diagrams, Reidemeister moves, bracket polynomial, Jones polynomial, periodic tangles, thickened torus, low-dimensional topology, periodic lattice, periodic boundary conditions, interdisciplinary applications |
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履修の際のアドバイス
Advice | | | This course is accessible to students with a general mathematical background, but familiarity with basic topology and algebra will be helpful. Active engagement with the suggested exercises and discussions will enhance understanding. Students are also encouraged to consider extensions of the concepts in other mathematical settings and to explore interdisciplinary applications and reach out for questions outside class hours as needed. |
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授業開講形態等
Lecture format, etc. | | | The course will be conducted in person through six 90-minute lectures over one week. Students will have access to lecture materials and optional exercises for independent study. |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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