Forms of surfaces spanned by a knot and behaviors of invariants

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03296
• 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: Nagoya Institute of Technology
• Principal Investigator: Hirasawa Mikami 名古屋工業大学, 工学(系)研究科(研究院), 教授 (00337908)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 結び目 / 絡み目不変量 / 多項式不変量 / ファイバー曲面 / アレクサンダー多項式 / 零点の分布

Outline of Final Research Achievements

We construct spanning surfaces for knots and examined the effect of their forms and deformations to the invariants of the knots. The shape of surfaces typically affects the distribution of the zeros of Alexander polynomials. We constructed explicit counter examples for the Hoste conjecture for alternating knots. Also, we recognized the phenomenon of the distribution of zeros being controlled by the twists in arborescent links. Study of the matrices of fiber surfaces lead us to the systematic construction of Salem polynomials. We generalized Stallings and Harer twists on fiber surfaces and examined their effect to the polynomials. We managed a constructive description of Seifert surfaces which are preserved by strong inversion of 2-bridge knots, and showed they realize the minimal equivariant genera. For that purpose we established a general method of showing the genus-minimality.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

アレクサンダー多項式は結び目の性質を顕著に表し，また補空間の構造ともよく馴染む，大変興味深い対象である．ホステ予想により，係数よりもむしろ零点の分布にも意義があると認識されてきた．絡み目に拡張してアレクサンダー多項式の零点を調べる中で，サーレム多項式の新しい構成法に気づいた．サーレム多項式は整数論や代数幾何でも重要な対象であり，結び目との関連付けを行えた．強対合で不変なザイフェルト曲面の研究は３，４次元において近年盛んに行われており，研究手法を貢献できたことにも意義がある．また古典的なファイバー曲面の研究にも新たな視覚化などで貢献できた．