Structures of knots and quandle cocycle invariants

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03476
• 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: Tsuda University
• Principal Investigator: Inoue Ayumu 津田塾大学, 学芸学部, 准教授 (10610149)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 結び目 / カンドル / ファイバー結び目 / 正多胞体 / ザイフェルト曲面 / 空間グラフ

Outline of Final Research Achievements

In mathematics, a knot means a knotted circle in the 3-dimensional space. A quandle is an algebraic system. For each quandle and each knot, we have quandle cocycle invariants of the knot, which are algebraic quantities for the knot.
Each knot is equipped with various kinds of structures, which characterize the knot. It is known experientially that quandle cocycle invariants of a knot detect several structures of the knot. The aim of this research is to investigate how or why quandle cocycle invariants detect structures of a knot. In accordance with this program, the researcher respectively figures out a relationship between a quandle cocycle invariant and "fiber structure", "twist-spinning structure" or "Seifert surface structure" of a knot.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

紐状の物質の絡まりは，例えば高分子や DNA など，自然界に多く存在する．その絡まりが備える「構造」を理解することは，物性や現象を理解する上においても，非常に重要である．本研究を通じて結び目が備える構造に対する理解が進んだことは，結び目理論の発展のみならず，科学全般の進展においても意義があると言える．