THE RESEARCH OF THE STRUCTURES AND INVARIANTS OF STABLE EQUIVALENCE CLASSES OF KNOTS IN THICKENED SURFACES

2012 Fiscal Year Final Research Report

• Project/Area Number: 22540093
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya City University
• Principal Investigator: KAMADA Naoko 名古屋市立大学, その他の研究科, 教授 (60419687)
• Co-Investigator(Renkei-kenkyūsha): KANENOBU Taizo 大阪市立大学, 理学(系)研究科(研究院), 教授 (00152819) ; SATOH Shin 神戸大学, 理学(系)研究科(研究院), 准教授 (90345009) ; MIYAZAWA Yasuyuki 山口大学, 理工学研究科, 教授 (60263761)
• Project Period (FY): 2010 – 2012
• Keywords: 位相幾何

Research Abstract

We defined the quandles of the stable equivalence classes of knots in thickened surfaces (twisted knots) and gave the geometric interpretation of the quandles of twisted knots. We also introduced the multivariable polynomial invariants and the index polynomials of twisted knots and showed that they are useful to distinguish twisted knots. We coded the computer programs to makes the list of twisted knot diagrams and to calculate the invariants of twisted knots. Using the programs, we constructed thetable of twisted knots.

Research Products

• [Journal Article] Biquandles with structures related to virtual links and twisted links (2012)
  • Author(s): 1 Naoko Kamada and Seiichi Kamada
  • Journal Title: J. Knot Theory Ramifications Volume: 21巻 Pages: 1240006 1-14
  • DOI: DOI:10.1142/S0218216512400068
  • Peer Reviewed
• [Journal Article] Polynomial invariants and quandles of twisted links (2012)
  • Author(s): Naoko Kamada
  • Journal Title: TopologyAppl. Volume: 159巻 Pages: 999-1006
  • DOI: DOI:10.1016/j.topol.2011.11.024
  • Peer Reviewed
• [Journal Article] Bridge presentations of virtual knots (2011)
  • Author(s): Mikami Hirasawa, Naoko Kamada, Seiichi Kamada
  • Journal Title: J. Knot Theory Ramifications Volume: 20巻 Pages: 881-893
  • DOI: DOI:10.1142/S0218216511009017
  • Peer Reviewed
• [Journal Article] New invariants of long virtual knots (2010)
  • Author(s): 4 Andrew Bartholomew, Roger Fenn, Naoko Kamada, Seiichi Kamada
  • Journal Title: Kobe J. Math Volume: 27巻 Pages: 21-33
  • Peer Reviewed
• [Journal Article] The polynomial invariants of twisted links (2010)
  • Author(s): Naoko Kamada
  • Journal Title: Topology Appl. Volume: 157巻 Pages: 220-227
  • Peer Reviewed
• [Journal Article] The Miyazawa polynomial for long virtual knots (2010)
  • Author(s): Atsushi Ishii, Naoko Kamada, SeiichiKamada
  • Journal Title: Topology Appl Volume: 157巻 Pages: 290-297
  • Peer Reviewed
• [Presentation] A surface bracket polynomial based on a multivariable polynomial invariant (2013)
  • Author(s): Naoko Kamada
  • Organizer: E-KOOK セミナー
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2013-02-15
• [Presentation] The surface state for the Miyazawa polynomial (2013)
  • Author(s): Naoko Kamada
  • Organizer: The 9th East Asian School of Knots and Related Topics
  • Place of Presentation: 東京大学
  • Year and Date: 2013-01-16
• [Presentation] サーフェイスブラケット多項式 (2012)
  • Author(s): Naoko Kamada
  • Organizer: 2012 琉球結び目セミナー
  • Place of Presentation: 那覇市伝統工芸館会議室
  • Year and Date: 2012-09-04
• [Presentation] Polynomial invariants and surface bracket polynomials of twisted links, Mini-symposium and special session STS "Knot Theory and its ramifications" (2012)
  • Author(s): Naoko Kamada
  • Organizer: The 6th European Congress of Mathematics
  • Place of Presentation: Krakow (Cracow), Poland
  • Year and Date: 2012-07-05
• [Presentation] Surface bracket polynomials of twisted links (2012)
  • Author(s): Naoko Kamada
  • Organizer: Friday Seminar on Knot Theory
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2012-06-22
• [Presentation] Twisted biquandles and invariants of twisted links (2012)
  • Author(s): Seiichi Kamada
  • Organizer: 国際会議「The 8th East Asian School of Knots and Related Topics」
  • Place of Presentation: KAIST, Korea
  • Year and Date: 2012-01-10
• [Presentation] Index polynomial invariants of virtual knots and twisted knots (2012)
  • Author(s): Naoko Kamada
  • Organizer: 国際会議「The 8th East Asian School of Knots and Related Topics」
  • Place of Presentation: KAIST, Korea
  • Year and Date: 2012-01-09
• [Presentation] Index polynomials of twisted links (2011)
  • Author(s): Naoko Kamada
  • Organizer: Quantum Topology Seminar / Knots in Chicago
  • Place of Presentation: The University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science, Chicago, USA
  • Year and Date: 2011-10-04
• [Presentation] Virtual knots as double covering of twisted knots (2011)
  • Author(s): Seiichi Kamada
  • Organizer: 国際会議「The 19th TAPU Seminar on Knots and Related Topics」
  • Place of Presentation: Merak Room, Seacloud Hotel (13F) Haeundae, Busan, Korea
  • Year and Date: 2011-09-16
• [Presentation] Biquandles with structures related to virtual links and twisted links (2011)
  • Author(s): Seiichi Kamada
  • Organizer: 国際会議「The 19th TAPU Seminar on Knots and Related Topics」
  • Place of Presentation: Merak Room, Seacloud Hotel (13F) Haeundae, Busan, Korea
  • Year and Date: 2011-09-15
• [Presentation] An interpretation of index polynomials (2011)
  • Author(s): Naoko Kamada
  • Organizer: 国際会議「The 19th TAPU Seminar on Knots and Related Topics」
  • Place of Presentation: Merak Room, Seacloud Hotel (13F) Haeundae, Busan, Korea
  • Year and Date: 2011-09-15
• [Presentation] Index 多項式について (2011)
  • Author(s): Naoko Kamada
  • Organizer: 2011 琉球結び目セミナー
  • Place of Presentation: 那覇市伝統工芸館会議室
  • Year and Date: 2011-09-06
• [Presentation] On checkerboard colorable twisted links (2011)
  • Author(s): Naoko Kamada
  • Organizer: 国際会議「 The Seventh East Asian School of Knots and Related Topics」
  • Place of Presentation: 広島大学
  • Year and Date: 2011-01-10
• [Presentation] Polynomial invariants and quandles of twisted links (2010)
  • Author(s): Naoko Kamada
  • Organizer: 国際会議「 International Conference Japan-Mexico on Topology and its Applications」
  • Place of Presentation: Universidad de Colima, Colima, Mexico
  • Year and Date: 2010-09-30
• [Presentation] On invariants of twisted links and applications (2010)
  • Author(s): Naoko Kamada
  • Organizer: 研究集会「The 2nd TAPU-KOOK Joint Seminar on Knot Theory and Related Topics」
  • Place of Presentation: Kyungpook National University, Korea
  • Year and Date: 2010-07-26
• [Presentation] On quandles and biquandles related to twisted virtual links (2010)
  • Author(s): Seiichi Kamada
  • Organizer: 研究集会「Knots in Washington XXX; Categorification, Quantum knots and Quantum computing」
  • Place of Presentation: George Washington University, Washington, DC, USA
  • Year and Date: 2010-05-20
• [Presentation] Quandles and multivariable polynomial invariants of twisted links (2010)
  • Author(s): Naoko Kamada
  • Organizer: 研究集会「 Knots in Washington XXX; Categorification, Quantum knots and Quantum computing」
  • Place of Presentation: George Washington University, Washington, DC, USA
  • Year and Date: 2010-05-20
• [Remarks]
  • URL: http://www.nsc.nagoya-cu.ac.jp/~kamada/tbltknot.html