2023 Fiscal Year Final Research Report

Geometric and algebraic aspects of Dehn surgery

Research Project

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Project/Area Number	19K03502
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	Nihon University
Principal Investigator	MOTEGI Kimihiko 日本大学, 文理学部, 教授 (40219978)
Co-Investigator(Kenkyū-buntansha)	新國亮東京女子大学, 現代教養学部, 教授 (00401878)
Project Period (FY)	2019-04-01 – 2024-03-31
Keywords	Dehn surgery / Dehn filling / L-space knot / twisting / tight fibered knot / generalized torsion / slope conjecture / strong slope conjecture
Outline of Final Research Achievements	We prove that any satellite L-space knot is braided, which answers a question of Baker-Moore and Hom in the positive. As an application we show that a satellite L-space knot cannot admit an essential Conway sphere, which gives a partial answer to a conjecture due to Lidman-Moore. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This answers a classical problem in group theory and gives a necessary and sufficient condition for the fundamental group of non-prime 3-manifold to have a generalized torsion element. We also demonstrate that the behavior of generalized torsion elements under torus decomposition is quite different from that under prime decomposition. Furthermore, we clarify how generalized torsion element arises via Dehn filling. Besides, we prove the strong slope conjecture for some satellite knots.
Free Research Field	低次元トポロジー
Academic Significance and Societal Importance of the Research Achievements	低次元トポロジーで近年注目されているL-空間結び目とねじり操作の関係を明らかにするとともに、接触幾何学の視点からブレイド軸を特徴づけることに成功した。その応用として、L-空間結び目に関するいくつかの未解決問題、予想を肯定的に解決することで当該分野の発展に貢献した。また、群論で重要なねじれ元の一般化である共役ねじれ元に関する未解決問題を解決することで、3次元多様体の素分解やトーラス分解のもとでの共役ねじれ元の振る舞いを明かにし、共役ねじれ元に焦点を当てた３次元トポロジーの研究の基礎を構築することができた。さらに、ストロングスロープ予想の部分的解決により量子トポロジーの発展に寄与した。