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

Extended knot

Research Project

Project/Area Number 15K04879
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionNagoya City University

Principal Investigator

Kamada Naoko  名古屋市立大学, 大学院システム自然科学研究科, 教授 (60419687)

Research Collaborator ISHII atsushi  
KAMADA seiichi  
KANENOBU taizo  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords結び目 / 仮想結び目 / 不変量
Outline of Final Research Achievements

We studied twisted knots, virtual knots, virtual doodle. Virtual knots correspond to stable equivalence classes of knots in thickened oriented closed surfaces. Twisted knots correspond to stable equivalence classes of knots in thickened closed surfaces. For some research methods of virtual knots, it is not easy to extend to those of twisted knots. We constructed the map from the set of twisted knots to that of virtual knots. As an application of this map, we defined an invariant of twisted knots and showed some applications. By extending this method, we introduced two normalization maps from a set of virtual knots to that of normal virtual knots. We showed some applications of them. We defined quandle invariants of virtual doodles by use of one of normalization maps. By extending one of normalization maps, we introduced the map from a set of virtual knots to that of mod m almost classical virtual knots. We showed some applications of this map.

Academic Significance and Societal Importance of the Research Achievements

拡張結び目には結び目を代数的に表す様々な方法に視点をおいて定義されたものもある。結び目不変量の中にはそれらの表し方と関係している場合もあり広い観点からのそれらの構造や性質の理解につながる。その中にはカンドルなどの代数構造をもつ不変量もあり代数への応用も期待される。また、拡張結び目と曲面結び目の関係はよく知られている。結び目不変量で拡張結び目の不変量として解釈されるものもある。このような視点から結び目の新しい研究手法を導入できる可能性があり結び目の分類にも寄与すると考えられる。

Report

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

    (31 results)

All 2019 2018 2017 2016 2015

All Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results,  Open Access: 2 results) Presentation (24 results) (of which Int'l Joint Research: 14 results,  Invited: 18 results) Funded Workshop (1 results)

  • [Journal Article] Doodles on Surfaces2018

    • Author(s)
      Bartholomew Andrew、Fenn Roger、Kamada Naoko、Kamada Seiichi
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 27 Issue: 12 Pages: 1850071

    • DOI

      10.1142/s0218216518500712

    • NAID

      120006772698

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2019992

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On Gauss codes of virtual doodles2018

    • Author(s)
      Bartholomew Andrew、Fenn Roger、Kamada Naoko、Kamada Seiichi
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 27 Issue: 11 Pages: 1843013

    • DOI

      10.1142/s0218216518430137

    • NAID

      120006764234

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2019991

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Coherent double coverings of virtual link diagrams2018

    • Author(s)
      Naoko Kamada
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 27 Issue: 11 Pages: 1843004-1843004

    • DOI

      10.1142/s0218216518430046

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A twisted link invariant derived from a virtual link invariant2017

    • Author(s)
      Kamada Naoko
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 26 Issue: 09 Pages: 1743007-1743007

    • DOI

      10.1142/s0218216517430076

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Converting virtual link diagrams to normal ones2017

    • Author(s)
      Kamada Naoko
    • Journal Title

      Topology and its Applications

      Volume: 230 Pages: 161-171

    • DOI

      10.1016/j.topol.2017.08.032

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Double coverings of twisted links2016

    • Author(s)
      Naoko Kamada and Seiichi Kamada
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 25 Issue: 9 Pages: 1641011

    • DOI

      10.1142/s021821651641011x

    • NAID

      120006528661

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2019990

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] Invariants and mod m almost classical virtual links2019

    • Author(s)
      Naoko Kamada
    • Organizer
      Algebraic and Combinatorial Structures in Knot Theory (Code: SS 9A) at the AMS Spring Central and Western Joint Sectional Meeting
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] A multivariable polynomial invariant of virtual links with cut systems2019

    • Author(s)
      Naoko Kamada
    • Organizer
      2019 KMJ Conference for Accreditation Strategies and the 2019 Winter TAPU Workshop on Knot Theory and Related Topics
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] A conversion of virtual links into mod p almost classical virtual links2019

    • Author(s)
      Naoko Kamada
    • Organizer
      The 14th East Asian Conference on Geometric Topology
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] 仮想結び目の多変数多項式不変量と oriented cut points2018

    • Author(s)
      Naoko Kamada
    • Organizer
      2018 年度琉球結び目セミナー
    • Related Report
      2018 Annual Research Report
  • [Presentation] Cyclic coverings of virtual knots2018

    • Author(s)
      Naoko Kamada
    • Organizer
      拡大KOOK セミナー2018
    • Related Report
      2018 Annual Research Report
  • [Presentation] Coverings of virtual links based on mod p Alexander numbering2018

    • Author(s)
      Naoko Kamada
    • Organizer
      The 10th KOOK-TAPU Joint Seminar on Knots and Related Topics
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Coherent double covering diagrams of virtual knots2018

    • Author(s)
      N. Kamada
    • Organizer
      国際会議「The 13th East Asian School of Knots and Related Topics」
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Coherent double covering diagrams of virtual links2018

    • Author(s)
      N. Kamada
    • Organizer
      国際会議「Knotted embeddings in dimensions 3 and 4」
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Virtual doodles and semiquandles2017

    • Author(s)
      Naoko Kamada, Andrew Bartholomew, Roger Fenn, Seiichi Kamada
    • Organizer
      国際会議「The 12th East Asian School of Knots and Related Topics」
    • Place of Presentation
      東京大学
    • Year and Date
      2017-02-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Virtual doodles and their invariants2017

    • Author(s)
      N. Kamada, A. Bartholomew, R Fenn, S. Kamada
    • Organizer
      国際会議「Self-distributive system and quandle (co)homology theory in algebra and low-dimensional topology 2017 KIAS Research Station Busan」
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Invariants of virtual doodles2017

    • Author(s)
      N. Kamada, A. Bartholomew, R Fenn, S. Kamada
    • Organizer
      国際会議「4th Russian-Chinese Conference on Knot Theory and Related Topics - 2017 」
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Double coverings of virtual doodles and their invariants2017

    • Author(s)
      N. Kamada, A. Bartholomew, R Fenn, S. Kamada
    • Organizer
      国際会議「The 2nd Pan Pacific International Conference on Topology and Applications 」
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The double covering method for twisted knots2017

    • Author(s)
      N. Kamada
    • Organizer
      「Intelligence of Low-dimensional Topology」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 仮想結び目から正規仮想結び目への写像2017

    • Author(s)
      N. Kamada
    • Organizer
      「東京女子大学トポロジーセミナー」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Virtual doodles and a quandle type invariant2016

    • Author(s)
      Naoko Kamada, Andrew Bartholomew, Roger Fenn, Seiichi Kamada
    • Organizer
      国際会議「KNOTS IN WASHINGTON XLIII」
    • Place of Presentation
      George Washington University, USA,
    • Year and Date
      2016-12-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Virtual doodle について2016

    • Author(s)
      鎌田直子
    • Organizer
      「拡大 KOOK セミナー 2016」
    • Place of Presentation
      大阪電気通信大学 寝屋川キャンパス
    • Year and Date
      2016-08-25
    • Related Report
      2016 Research-status Report
  • [Presentation] Checkerboard colorable virtual knot2016

    • Author(s)
      Naoko Kamada
    • Organizer
      国際会議「Asian Mathematical Conference 2016」
    • Place of Presentation
      Bali Nusa Dua Convention Center (BNDCC), Bali, Indonesia
    • Year and Date
      2016-07-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Converting virtual knot diagrams to normal diagrams2016

    • Author(s)
      Naoko Kamada
    • Organizer
      国際会議「International Workshop on Low-dimensional Topology」
    • Place of Presentation
      Dalian University of Technology, Dalian, China
    • Year and Date
      2016-05-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Converting a virtual link diagram to a normal one2016

    • Author(s)
      鎌田直子
    • Organizer
      International Conference, The 11th East Asian School of Knots and Related Topics
    • Place of Presentation
      大阪市立大学杉本キャンパス(大阪市住吉区)
    • Year and Date
      2016-01-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Normal virtual link diagrams and double covers of twisted link diagrams2016

    • Author(s)
      鎌田直子
    • Organizer
      JMM Joint Mathematics Meetings The Mathematical Association of America (MAA) and the American Mathematical Society (AMS)
    • Place of Presentation
      Seattle, USA
    • Year and Date
      2016-01-07
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Normal virtual link diagrams and double covers of twisted link diagrams2015

    • Author(s)
      鎌田直子
    • Organizer
      The 1st Pan Pacific International Conference on Topology and Applications
    • Place of Presentation
      Min Nan Normal University, Zhangzhou, China
    • Year and Date
      2015-11-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Virtual knot diagrams and double covers of twisted knot diagrams2015

    • Author(s)
      鎌田直子
    • Organizer
      TAPU Workshop 2015
    • Place of Presentation
      Jeju National University International Center, Korea
    • Year and Date
      2015-09-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted knot の不変量とdouble cover2015

    • Author(s)
      鎌田直子
    • Organizer
      拡大 KOOK セミナー 2015
    • Place of Presentation
      神戸大学六甲台第2キャンパス(神戸市灘区)
    • Year and Date
      2015-08-19
    • Related Report
      2015 Research-status Report
  • [Presentation] A generalization of the JKSS invariant to twisted knots2015

    • Author(s)
      鎌田直子
    • Organizer
      Low Dimensional Topology and Its Relationships with Physics, AMS/EMS/SPM Meetings
    • Place of Presentation
      Porto, Portugal
    • Year and Date
      2015-06-13
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Knotting Nagoya International mini-workshop 名古屋工業大学(愛知県名古屋市)2017

    • Related Report
      2017 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi