Study of knots using local moves

2023 Fiscal Year Final Research Report

• Project/Area Number: 16K05162
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka Institute of Technology
• Principal Investigator: Tsukamoto Tatsuya 大阪工業大学, 工学部, 教授 (10350480)
• Project Period (FY): 2016-04-01 – 2024-03-31
• Keywords: 結び目理論

Outline of Final Research Achievements

The purpose of this research is to know the structure of the set of whole knots in the 3-space and the topological property of each knot by using local moves. In the term of research (2016-2023), we worked on simple-ribbon fusions and pretzel knots and obtained several results. In general, it is hard to calculate the value of knot invariants and difference between the before and after knots when we apply local moves. However, we calculated the difference of Alexander polynomials for the case of simple-ribbon fusions, and the values of Alexander polynomial of pretzel knots whose parameter sequences are erasable. Moreover, using the results, we determined simple-ribbon knots whose crossing number is less than equal to ten, and simple-ribbon knots which are odd stranded even pretzel.

Free Research Field

低次元トポロジー

Academic Significance and Societal Importance of the Research Achievements

研究対象である結び目は３次元多様体や整数論といった数学の分野だけではなく、DNA研究のような数学外の分野とも深く関連している。実際、特に注力している局所変形の研究は組み換え酵素によるDNAへの作用に対応している。そのような中、本研究では単純リボン融合でほどける結び目のアレキサンダー多項式や、可約性をもつプレッツェル結び目のアレキサンダー多項式を求めた。さらにスライス・リボン予想という結び目理論における大きな予想の１つに対し、部分解を与えた。