• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Forms of surfaces spanned by a knot and behaviors of invariants

Research Project

Project/Area Number 18K03296
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionNagoya 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
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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 / ザイフェルト曲面 / 多項式の零点 / 絡み目 / 組み紐 / 結び目不変量 / 強対合の対称性 / 2橋結び目 / 村杉和 / 零点 / knot / Seifert surface / alexander polynomial / 曲面 / 不変量
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.

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.

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (37 results)

All 2024 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (15 results) Journal Article (7 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results,  Open Access: 2 results) Presentation (12 results) (of which Int'l Joint Research: 6 results,  Invited: 7 results) Remarks (3 results)

  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Instituto de Ciencias Matematicas(スペイン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Genvea(スイス)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] ジュネーブ大学(スイス)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] ミシガン州立大学(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] クラーク大学(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] ジュネーブ大学(スイス)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] ミシガン州立大学(米国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ジュネーブ大学(スイス)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Clerk University(米国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Toronto(カナダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Univ. of Toronto(カナダ)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Saddle point braids of braided fibrations and pseudo-fibrations2024

    • Author(s)
      B. Bode and M. Hirasawa
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 11 Issue: 2 Pages: 1-39

    • DOI

      10.1007/s40687-024-00446-x

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The equivariant genera of marked strongly invertible knots associated with 2-bridge knots2024

    • Author(s)
      M. Hirasawa, R. Hiura and M. Sakuma
    • Journal Title

      Michigan Mathematical Journal.

      Volume: -

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Construction and manipulation of Seifert surfaces in knot theory (a note in 2023)2024

    • Author(s)
      M. Hirasawa
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2263 Pages: 21-38

    • Related Report
      2023 Annual Research Report
    • Open Access
  • [Journal Article] Invariant Seifert surfaces for strongly invertible knots2023

    • Author(s)
      M. Hirasawa, R. Hiura and M. Sakuma
    • Journal Title

      IRMA Lect. Math. Theor. Phys.

      Volume: 34 Pages: 325-349

    • DOI

      10.4171/irma/34/16

    • ISBN
      9783985470242, 9783985475247
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Invariant Seifert surfaces for strongly invertible knots2023

    • Author(s)
      M. Hirasawa, R. Hiura and M. Sakuma
    • Journal Title

      Essays in Geometry - Dedicated to Norbert A'Campo.

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stable Alexander polynomials of arborescent links2020

    • Author(s)
      M. Hirasawa and K. Murasugi
    • Journal Title

      Jour. Knot Theory and its Ramifications

      Volume: 28 Issue: 13 Pages: 1940017-1940017

    • DOI

      10.1142/s0218216519400170

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Alternating knots with Alexander polynomials having unexpected zeros2018

    • Author(s)
      M.Hirasawa, K.Ishikawa and M.Suzuki
    • Journal Title

      Topology and its applications

      Volume: 253 Pages: 48-56

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] The equivariant genera of marked strongly invertible knots associated with 2-bridge knots2024

    • Author(s)
      M. Hirasawa
    • Organizer
      SEMINAIRE DE TOPOLOGIE ET GEOMETRIE
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction and manipulation of Seifert surfaces in knot theory2023

    • Author(s)
      M. Hirasawa
    • Organizer
      Intelligence of Low-dimensional Topology
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stallings’, Harer’s and one more, last twist on fiber surfaces producing new fibered links2023

    • Author(s)
      M. Hirasawa
    • Organizer
      18th East Asian Conference on Geometric Topology
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] 2橋結び目の強対合で不変なザイフェルト曲面の最小種数2022

    • Author(s)
      M. Hirasawa
    • Organizer
      拡大KOOKセミナー2022
    • Related Report
      2022 Research-status Report
  • [Presentation] Sphere eversion in virtual reality2022

    • Author(s)
      Mikami Hirasawa
    • Organizer
      The 17th East Asian Conference on Geometric Topology
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] On Alexander polynomials of degree over 764 of alternating knots with over 778 crossings2021

    • Author(s)
      Mikami Hirasawa
    • Organizer
      Mini-Workshop "Knots + More"
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 有理数結び目について2019

    • Author(s)
      M.Hirasawa
    • Organizer
      第 41 回愛知数論セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Fiberation of knot complements and partial chirality of knots2019

    • Author(s)
      M.Hirasawa
    • Organizer
      キラリティー、トポロジー、結び目論 第3回研究会
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Alternating knots with Alexander polynomials having unexpected zeros2019

    • Author(s)
      M.Hirasawa
    • Organizer
      The 14th East Asian Conference on Geometric Topology
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On the Distribution of Zeros of Alexander Polynomials of links2018

    • Author(s)
      M.Hirasawa
    • Organizer
      SMINAIRE DE TOPOLOGIE ET GEOMETRIE
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Alternating knots with Alexander polynomials having unexpected zeros2018

    • Author(s)
      M.Hirasawa
    • Organizer
      トポロジーとコンピュータ
    • Related Report
      2018 Research-status Report
  • [Presentation] Some symmetry-preserving crossing changes from homologically fibered knots to fibered knots2018

    • Author(s)
      M.Hirasawa
    • Organizer
      琉球結び目セミナー
    • Related Report
      2018 Research-status Report
  • [Remarks] 名古屋工業大学研究者データベースシステム

    • URL

      https://researcher.nitech.ac.jp/html/100000100_ja.html

    • Related Report
      2023 Annual Research Report
  • [Remarks] 名古屋工業大学 研究者詳細

    • URL

      https://researcher.nitech.ac.jp/html/100000100_ja.html

    • Related Report
      2022 Research-status Report
  • [Remarks] 名古屋工業大学 研究者データベースシステム

    • URL

      http://researcher.nitech.ac.jp/html/100000100_ja.html

    • Related Report
      2020 Research-status Report 2019 Research-status Report

URL: 

Published: 2018-04-23   Modified: 2025-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi