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A study on knots and mapping class groups using Heegaard Floer theory

Research Project

Project/Area Number 15K04865
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionYamagata University

Principal Investigator

Matsuda Hiroshi  山形大学, 理学部, 准教授 (70372703)

Project Period (FY) 2015-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords結び目 / ホモロジー / 横断的結び目 / ホモロジー群 / コード代数 / 接触ホモロジー
Outline of Final Research Achievements

There are two methods of representing 2-dimensional knots in the 4-sphere, using marked graphs in the 3-sphere, and using projections into the 3-sphere. I constructed homological invariants for 2-dimensional knots in the 4-sphere using these two representations. Calculating these homologies explicitly, it was shown that both distinguish the 0-twist spun-trefoil knot from the 2-twist spun-trefoil knot.

Academic Significance and Societal Importance of the Research Achievements

1次元結び目の研究においては 結び目図式のスケイン関係式を用いて多項式不変量やホモロジー不変量の研究が発展している。しかし2次元結び目の研究においては結び目図式のスケイン関係式をうまく定義することができないため基本群やカンドル構造を使った研究が中心である。1次元結び目の研究においてスケイン関係式を使わずに定義された接触ホモロジー理論から着想を得て、2次元結び目の研究にホモロジー不変量を導入することができた。これにより2次元結び目の研究においても1次元結び目の研究に追随する進展を期待することができる。

Report

(7 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (6 results)

All 2020 2019 2017 2015

All Journal Article (1 results) (of which Peer Reviewed: 1 results) Presentation (5 results) (of which Int'l Joint Research: 1 results,  Invited: 3 results)

  • [Journal Article] 2-knot homology and Yoshikawa move2020

    • Author(s)
      Hiroshi Matsuda
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 29 Issue: 09 Pages: 2050067-2050067

    • DOI

      10.1142/s0218216520500674

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Presentation] 2-knot homologies: Roseman and Yoshikawa2020

    • Author(s)
      松田 浩
    • Organizer
      接触構造、特異点、微分方程式及びその周辺
    • Related Report
      2019 Research-status Report
  • [Presentation] 2-knot homologies: Roseman and Yoshikawa2019

    • Author(s)
      松田 浩
    • Organizer
      拡大KOOKセミナー2019
    • Related Report
      2019 Research-status Report
  • [Presentation] 境界付き多様体のHeegaard Floer理論2017

    • Author(s)
      松田 浩
    • Organizer
      接触構造、特異点、微分方程式及びその周辺
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県金沢市)
    • Year and Date
      2017-01-19
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Homological invariants of surface-knots2017

    • Author(s)
      Hiroshi Matsuda
    • Organizer
      Differential Topology 17
    • Place of Presentation
      電気通信大学(東京都調布市)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Bordered Floer homology of Torelli elements2015

    • Author(s)
      松田 浩
    • Organizer
      リーマン面に関連する位相幾何学
    • Place of Presentation
      東京大学大学院数理科学研究科
    • Year and Date
      2015-08-26
    • Related Report
      2015 Research-status Report
    • Invited

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Published: 2015-04-16   Modified: 2022-01-27  

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