Symmetric unions and essential surfaces

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K03465
• 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: Gifu University
• Principal Investigator: Tanaka Toshifumi 岐阜大学, 教育学部, 准教授 (60396851)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: リボン結び目 / 対称和 / 曲面

Outline of Final Research Achievements

The purpose of this study is to characterize and classify symmetric unions. In 1984, Menasco introduced an effective method for studying a surface in the exterior of an alternating knot and solved some important problems in knot theory. In that method, a surface is characterized by considering the property of curves that appears at the intersection of the surface and a sphere of the knot complement. I have characterized and classified symmetric unions by investigating the intersection of surfaces in the knot complement using similar methods. As a result, we could characterize surfaces in the knot complement using the three-dimensional topology and characterize symmetric unions

Free Research Field

トポロジー

Academic Significance and Societal Importance of the Research Achievements

これまでの結び目の対称和の研究では，主に代数的な手法が用いられてきた。私は本研究において，3次元位相幾何学的に手法を用いて研究を行い，実際にそれが対称和の分類に役立つことを示す研究成果を得ることができた。さらにその幾何学的手法を発展させることで，本研究分野において，より多くの研究成果を得られることが期待できる。