Development and application of keen geodesics in the curve complex

2023 Fiscal Year Final Research Report

• Project/Area Number: 22K20335
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Aichi University of Education
• Principal Investigator: Ido Ayako 愛知教育大学, 教育学部, 准教授 (00759532)
• Project Period (FY): 2022-08-31 – 2024-03-31
• Keywords: Heegaard分解 / 曲線複体 / 橋分解 / Hempel距離

Outline of Final Research Achievements

We studied the concept "keen" of the curve complex, and researched on Hempel distance for Heegaard splittings of 3-manifolds and bridge splittings of links. Then we showed the existence of keen bridge splittings of links with Hempel distance 1, and proved that keen cannot be realized in some cases. Furthermore, we found that the construction methods of keen bridge splittings can be successfully applied to the construction of "weakly keen" bridge splittings with Hempel distance 1.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた成果からweakly keenな3次元多様体の実現が期待できるようになった．これらはGoeritz群の有限性など3次元多様体の研究に大きく貢献できると考えられる．また，keenという性質がGoeritz群の有限性を判別する手法となっていることからもわかるように，本研究の成果は，曲線複体の諸性質の汎用性を今後さらに高めることに寄与するものと考えられる．