Extended knots and their invariants

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03496
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Nagoya City University
• Principal Investigator: Kamada Naoko 名古屋市立大学, 大学院理学研究科, 教授 (60419687)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 結び目 / 拡張結び目 / 不変量

Outline of Final Research Achievements

Extended knots are extensions of knots, but they exhibit various differences in geometric properties, which can sometimes prevent the natural utilization or extension of conventional research methods for knots, including knot invariants. In this study, we investigated the geometric properties of extended knots (virtual knots, twisted knots, and welded knots), and obtained results. Furthermore, we introduced invariants and classification methods for virtual knots, twisted knots, and others, and showed examples of their applications.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

結び目理論はトポロジーの一分野であり、拡張結び目は結び目を様々な視点で拡張した概念であり、その視点には幾何学的特徴や結び目不変量などがある。幾何学的特徴や結び目不変量を広い視点で調べることはその構造や性質の理解につながる。結び目不変量は代数系への写像であり代数学への応用が考えられる。本研究での拡張結び目の幾何学的側面や不変量に関する成果は、トポロジーや代数学への寄与が期待できる。