On the complement conjecture for knots in lens spaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03287
• 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: ICHIHARA Kazuhiro 日本大学, 文理学部, 教授 (00388357)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 結び目 / レンズ空間 / デーン手術

Outline of Final Research Achievements

The main subject of this research is to consider the problem asking when the complement of a knot determines the type of the knot. This is one of the fundamental problems in Knot theory. The problem was solved for knots in the 3-sphere in the late 1980's. In this research, we focused on knots in lens spaces, which give a simple class of 3-manifolds including the 3-sphere. For the problem, to study the operation to create a 3-manifold, called Dehn surgery, has played quite an important role. In fact, the main part of this research is focused on cosmetic Dehn surgeries, which generate homeomorphic manifold pairs. As a research result, several partial solutions to this problem were obtained. In addition, the fact that this research has led to new advances in research both in Japan and overseas can be regarded as an indirect result of our research.

Free Research Field

数物系科学

Academic Significance and Societal Importance of the Research Achievements

空間内の結び目を数学的に研究する際，その補空間に着目することが多くなされている。実際，結び目が同値（連続変形でうつりあう）ならば補空間は同相（位相幾何において等しい）ということが容易にわかる。しかし，その逆，補空間が同相ならば結び目が同値になるか？という問題は自明でなく，長い間，未解決問題であった。1980年代にこの問題は最も基礎的な3次元球面内の結び目については肯定的に解決されたが，一般の3次元多様体内の結び目については現在も未解決である。本研究では，レンズ空間と呼ばれるクラスの3次元多様内の結び目についてこの問題を研究し，いくつかの部分的解決を得た。