表題番号：2022C-434
日付：2023/03/31
研究課題代数系を用いたルジャンドル結び目の新しい不変量の構成
| 研究者所属（当時） | 資格 | 氏名 | |
|---|---|---|---|
| （代表者） | 理工学術院 基幹理工学部 | 助手 | 木村 直記 |
| （連携研究者） | 津田塾大学 | 准教授 | 井上 歩 |
| （連携研究者） | 東京女子大学 | 教授 | 新國 亮 |
| （連携研究者） | 早稲田大学 | 教授 | 谷山 公規 |
- 研究成果概要
- The purpose of this research is to construct a new invariant of Legendrian knots in contact 3-manifolds and to investigate its properties. The study on Legendrian knots plays an important role in 3-dimensional contact topology.
I defined new invariants of Legendrian knots using rack colorings and proved that those invariants can distinguish Legendrian unknots.
A paper on the results titled “Bi-Legendrian rack colorings of Legendrian knots” is to appear in Journal of Knot Theory and Its Ramifications.
Another theme in this research is a Legendrian embedding of a graph into the 3-space as a generalization of Legendrian knots. Ayumu Inoue, Ryo Nikkuni, Kouki Taniyama and I proved that for a graph which satisfies some conditions, the parity of the sum of crossing numbers of cycles in the graph immersed into the plane is independent of the choice of the immersion. As a corollary of the theorem, we showed that the Petersen graph and the Heawood graph have no Legendrian embeddings satisfying a certain condition.
A paper on the results titled “Crossing numbers and rotation numbers of cycles in a plane immersed graph” has published in Journal of Knot Theory and Its Ramifications.