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Diagrammatic research on similarity and difference between 1- and 2-dimensional knots

Research Project

Project/Area Number 16K05147
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

SATOH SHIN  神戸大学, 理学研究科, 教授 (90345009)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords曲面結び目 / 射影図 / 仮想結び目 / 溶接結び目 / 不変量 / ガウス図 / ケーブル結び目 / ねじれ多項式 / 指数多項式 / 局所変形 / 被覆結び目 / パス変形 / 2次元結び目 / 多重化 / 彩色 / トーラス結び目 / 二橋結び目 / パレット数 / 曲面タングル / リボン結び目 / ダブル / トポロジー
Outline of Final Research Achievements

(1) We determine the palette numbers of n-colorable torus knots and 2-bridge knots. (2) We introduce the double of a surface-knot via ribbon surface-tangles, and construct a natural map from the set of surface-knots to that of the stable classes of ribbon surface-knots. (3) We characterize the writhe polynomial of a virtual knot by shell moves, and extend it to the case of 2-component virtual links. (4) We prove that the pass move is an unknotting operation for welded knots. (4) We introduce a presentation of the double point set of a diagram of a surface-knot by using a Gauss diagram with trivalent chords, and prove that it recovers the Gauss diagram of the corresponding double decker set of the surface-knot diagram. In particular, we give several properties of the double point set in the case of a 2-dimensional knot.

Academic Significance and Societal Importance of the Research Achievements

本研究の課題は1次元および2次元の結び目の性質を、射影図という観点から明らかにすることであった。(1)これまで具体的なnの値でしか決定されていなかったnパレット数を大きなクラスで一般に決定できた。(2)カンドルや結び目群といった曲面結び目の研究がリボン曲面結び目に限定することができるようになった。(3)結び目の不変量と局所変形という、幾何と代数をつなぐ観点から、ねじれ多項式に対応する局所変形を見つけることができた意義は大きい。(4)結び目のアーフ不変量は溶接結び目に拡張できないことを意味している。(5)2次元結び目の2重点集合の性質により、3重点を用いたリスト作成に役立てることができる。

Report

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

    (23 results)

All 2020 2019 2018 2017 2016

All Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 10 results,  Open Access: 3 results,  Acknowledgement Compliant: 2 results) Presentation (13 results) (of which Int'l Joint Research: 3 results,  Invited: 11 results)

  • [Journal Article] Writhe polynomials and shell moves for virtual knots and links2020

    • Author(s)
      T. Nakamura, Y. Nakanishi, and S. Satoh
    • Journal Title

      European J. Combin.

      Volume: 84

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The pass move is an unknotting operation for welded knots2018

    • Author(s)
      T. Nakamura, Y. Nakanishi, S. Satoh, and A. Yasuhara
    • Journal Title

      Topology Appl.

      Volume: 247 Pages: 9-19

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Crossing changes, delta moves and sharp moves on welded knots2018

    • Author(s)
      S. Satoh
    • Journal Title

      Rocky Mountain J. Math.

      Volume: 48 Pages: 967-979

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Finiteness of the set of virtual knots with a given state number2018

    • Author(s)
      T. Nakamura, Y. Nakanishi, and S. Satoh
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 27

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] The palette numbers of torus knots2017

    • Author(s)
      T. Hayashi, T. Nakamura, Y. Nakanishi, and S. Satoh
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 26

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The palette numbers of 2-bridge knots2017

    • Author(s)
      T. Nakamura, Y. Nakanishi, M. Saito, and S. Satoh
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 26

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The 6- and 8-palette numbers of links2017

    • Author(s)
      T. Nakamura, Y. Nakanishi, M. Saito, and S. Satoh
    • Journal Title

      Topology App.

      Volume: 222 Pages: 200-216

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] 11-colored knot diagram with five colors2016

    • Author(s)
      T. Nakamura, Y. Nakanishi, and S. Satoh
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 25

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A note on the OU sequences of a 2-bridge knot2016

    • Author(s)
      Y. Funahashi, Y. Nakanishi, and S. Satoh
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 25

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The length of a 3-cocycle of the 5-dihedral quandle2016

    • Author(s)
      S. Satoh
    • Journal Title

      Algebra. Geom. Topol.

      Volume: 16 Pages: 3325-3359

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] Shell moves for 2-component virtual links2019

    • Author(s)
      佐藤進
    • Organizer
      2019日本数学会秋季総合分科会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Double point circles of regular 2-knot diagrams2018

    • Author(s)
      佐藤進
    • Organizer
      研究集会「拡大KOOKセミナー2018」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] The n-cable of a ribbon 2-knot2018

    • Author(s)
      S. Satoh
    • Organizer
      Four Dimensional Topology
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A property of regular 2-knot diagrams2018

    • Author(s)
      佐藤進
    • Organizer
      研究集会「2018年度琉球結び目セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 2次元結び目の多重化について2018

    • Author(s)
      佐藤進
    • Organizer
      2017年度琉球結び目セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On the double of a surface-link2017

    • Author(s)
      Shin Satoh
    • Organizer
      The 12th East Asian School of Knots and Related Topics
    • Place of Presentation
      東京大学
    • Year and Date
      2017-02-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A set of local moves generating the writhe polynomial2017

    • Author(s)
      佐藤進
    • Organizer
      拡大KOOKセミナー2017
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] リボン曲面タングルと曲面絡み目のダブル2017

    • Author(s)
      佐藤進
    • Organizer
      2017年度秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] 単純2次元結び目図式のOUグラフについて2016

    • Author(s)
      佐藤進
    • Organizer
      2016年度琉球結び目セミナー
    • Place of Presentation
      那覇市伝統工芸館
    • Year and Date
      2016-12-18
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] The ribbon stable class of a surface-link2016

    • Author(s)
      Shin Satoh
    • Organizer
      Friday Seminar on Knot Theory
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2016-12-02
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Fundamental deformations of unwrithed surface-knots2016

    • Author(s)
      佐藤進
    • Organizer
      4次元トポロジー
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2016-11-25
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A construction of stable classes of ribbon surface-knots from non-ribbon surface-knots2016

    • Author(s)
      Shin Satoh
    • Organizer
      KOOK-TAPU Workshop of Knots in Tsushima Island
    • Place of Presentation
      対馬市交流センター
    • Year and Date
      2016-09-08
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 曲面結び目のねじれのない射影図2016

    • Author(s)
      佐藤進
    • Organizer
      拡大KOOKセミナー
    • Place of Presentation
      大阪電気通信大学
    • Year and Date
      2016-08-22
    • Related Report
      2016 Research-status Report
    • Invited

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Published: 2016-04-21   Modified: 2021-02-19  

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