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STUDY OF TOPOLOGICAL INVARIANTS FOR KNOTS AND THEIR APPLICATIONS

Research Project

Project/Area Number 14340027
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka City University

Principal Investigator

KANENOBU Taizo  Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00152819)

Co-Investigator(Kenkyū-buntansha) KAWAUCHI Akio  Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
KOMORI Youhei  Osaka City Univ., Graduate School of Science, Ass.Professor, 大学院・理学研究科, 助教授 (70264794)
KAMADA Seiichi  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60254380)
NAKANISHI Yasutaka  Kobe University, Faculty of Science, Professor, 理学部, 教授 (70183514)
OHYAMA Yoshiyuki  Tokyo Women's Christian Univ., College of Arts and Science, Prof., 理学部, 教授 (80223981)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥8,100,000 (Direct Cost: ¥8,100,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2002: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsknot / link / Cn-move / finite type invariant / HOMFLY polynomial / Links-Gould polynomial / Kauffman polynomial / virtual knot / スケイン関係式 / 2変数アレキサンダー多項式 / 2本組み紐仮想結び目 / 仮想結び目群 / リボントーラス結び目 / 2次元球面結び目 / 交換子部分群 / リンクス-グールド不変量 / 2本橋絡み目 / 両手性 / 係数多項式 / デルタ変形 / C n変形 / 有限型不変量 / Conway多項式 / HOMFLY多項式
Research Abstract

There are five main results.
1.We investigated how a delta move influences the first HOMFLY coefficient polynomials of a link. Then we generalized this to a Cn-move.
2.We studied the Links-Gould (LG) polynomial, which is a quantum invariant. Using a skein relation discovered by Ishii, we found a way to construct knots or links sharing the same LG polynomial. Then we gave arbitrarily many 2-bridge knots and links with the same LG polynomial. These 2-bridge knots and links also share the same HOMFLY, Kauffman, and 2-variable Alexander (in case of links) polynomials.
3.We give formulas for the first four coefficient polynomials of the Kauffman's link polynomial involving linking numbers and the coefficient polynomials of the Kauffman polynomials of the one- and two-component sublinks.
4.Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of certain special families of 2-braid virtual knots. It is known that the collection of the virtual knot groups is the same as that of the ribbon torus-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon 2-sphere-knot groups, 2-sphere-knot groups, torus-knot groups, and 3-sphere-knot groups.
5.We give a skein relation for the HOMFLYPT polynomials of 2 cable links. Using this, we have shown that the collection of 2-bridge knots or links mentioned in part 2 also share the same 2-cable HOMFLY polynomial.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (42 results)

All 2005 2004 2003 2002 Other

All Journal Article (28 results) Book (1 results) Publications (13 results)

  • [Journal Article] Delta move and polynomial invariants of links2005

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Topology and its Applications, Proceedings of the Second Joint Japan-Mexico Meeting in Topology (Matsue, 2002) 146-147

      Pages: 91-104

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Different links with the same Links-Gould invariant2005

    • Author(s)
      Ishii, Atsushi
    • Journal Title

      Osaka Journal of Mathematics 42・2

      Pages: 273-290

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Delta move and polynomial invariants of links.2005

    • Author(s)
      Kanenobu, Taizo, Nikkuni, Ryo
    • Journal Title

      Topology Appl. 146-147

      Pages: 91-104

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Different links with the same Links-Gould invariant.2005

    • Author(s)
      Ishii, Atsushi, Kanenobu, Taizo
    • Journal Title

      Osaka J.Math. 42 no.2

      Pages: 273-290

    • NAID

      120004846201

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Delta move and polynomial invariants of links2005

    • Author(s)
      Kanenobu, Taizo, Nikkuni, Ryo
    • Journal Title

      Topology and its Applications 146/147

      Pages: 91-104

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Different links with the same Links-Gould invariant2005

    • Author(s)
      Ishii, Atsushi, Kanenobu, Taizo
    • Journal Title

      Osaka J.Math. 42・2

      Pages: 273-290

    • NAID

      120004846201

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Delta move and polynomial invariants of links2005

    • Author(s)
      Kanenobu, Taizo, Nikkuni, Ryo
    • Journal Title

      Topology and its Applications 146(147)

      Pages: 91-104

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Topological imitation of a colored link with the same Dehn surgery manifold2005

    • Author(s)
      Kawauchi, Akio
    • Journal Title

      Topology and its Applications 146(147)

      Pages: 67-82

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A note on unknotting number, II2005

    • Author(s)
      Nakanishi, Yasutaka
    • Journal Title

      Journal of Knot Theory and Its Ramifications 14・1

      Pages: 3-8

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Enveloping monoidal quandles2005

    • Author(s)
      Kamada, Seiichi, Matsumoto, Yukio
    • Journal Title

      Topology and its Applications 146(147)

      Pages: 133-148

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Cn-moves and the HOMFLY polynomials of links2004

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Bol. Soc. Mat. Mexicana (3), Special Issue 10

      Pages: 263-277

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Cn - moves and the HOMFLY polynomials of links.2004

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Bol.Soc.Mat.Mexicana (3)10, Special Issue

      Pages: 263-277

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tangles with up to seven crossings2003

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Interdiscip. Inform. Sci. 9・1

      Pages: 127-140

    • NAID

      110000071785

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tangles with up to seven crossings.2003

    • Author(s)
      Kanenobu, Taizo, Saito, Hirofusa, Satoh, Shin
    • Journal Title

      Interdiscip.Inform.Sci. 9, no.1

      Pages: 127-140

    • NAID

      110000071785

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Two filtrations of ribbon 2-knots2002

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Topology and its Applications, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999) 121・1-2

      Pages: 143-168

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Virtual arc presentations and HC-moves of ribbon 2-knots2002

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Journal of Knot Theory and its Ramifications 11・3

      Pages: 387-402

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tangle surgeries on the double of a tangle and link polynomial2002

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Kobe Journal of Mathematics 19・1-2

      Pages: 1-19

    • NAID

      110001052863

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Two filtrations of ribbon 2-knots.2002

    • Author(s)
      Kanenobu, Taizo, Shima, Akiko
    • Journal Title

      Topology and its Applications, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), Topology Appl. 121, no.1-2

      Pages: 143-168

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Virtual arc presentations and HC-moves of ribbon 2-knots.2002

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      J.Knot Theory Ramifications 11 no.3

      Pages: 387-402

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tangle surgeries on the double of a tangle and link polynomials.2002

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Kobe J.Math. 19, no.1-2

      Pages: 1-19

    • NAID

      110001052863

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The first four terms of Kauffman's link polynomial

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Kyungpook Mathematical Journal

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A relation between the Links-Gould invariant and the Kauffman polynomial

    • Author(s)
      Ishii, Atsushi
    • Journal Title

      Topology Appl., Proceedings of JAMEX III (出版予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Groups of two-braid virtual knots

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Journal of Knot Theory and its Ramifications (出版予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The first four terms of Kauffman's link polynomial.

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Kyungpook Mathematical Journal (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A relation between the Links-Gould invariant and the Kauffman polynomial.

    • Author(s)
      Ishii, Atsushi, Kanenobu, Taizo
    • Journal Title

      Proceedings of JAMEX III, Topology Appl. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Groups of two-braid virtual knots.

    • Author(s)
      Kanenobu, Taizo, Tsuji, Kazunori
    • Journal Title

      J.Knot Theory Ramifications (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Cn - moves and the HOMFLY polynomials of links

    • Author(s)
      Kanenobu, Taizo
    • Journal Title

      Boletin de la Sociedad Matematica Mexicana (出版予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Different links with the same Links-Gould invariant

    • Author(s)
      Kanenobu, Taizo, Atsushi Ishii
    • Journal Title

      Osaka J.Math. (出版予定)

    • NAID

      120004846201

    • Related Report
      2004 Annual Research Report
  • [Book] 曲面・結び目・多様体のトポロジー2003

    • Author(s)
      S.C.カールソン著;金信泰造訳
    • Total Pages
      200
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] Kanenobu, Taizo, Saito, Hirofusa, Satoh, Shin: "Tangles with up to seven crossings"Interdiscip.Inform.Sci.. 9-1. 127-140 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kanenobu, Taizo, Nikkuni, Ryo: "Delta move and polynomial invariants of links"Top.Appl.Proceedings of the Second Joint Japan-Mexico Meeting in Topology (Matsue, 2002). (出版予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Kanenobu, Taizo: "Cn-moves and the HOMFLY polynomials of links"Boletin de la Sociedad Matematica Mexicana. (出版予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Kawauchi, Akio: "Topological imitation of a colored link with the same Dehn surgery manifold"Top.Appl.Proceedings of the Second Joint Japan-Mexico Meeting in Topology (Matsue, 2002). (出版予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Carter, J.Scott, Jelsovsky, Daniel, Kamada, Seiichi, Langford, Laurel, Saito, Masahico: "Quandle cohomology and state-sum invariants of knotted curves and surfaces"Trans.Amer.Math.Soc.. 355・10. 3947-3989 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Nakanishi, Yasutaka, Ohyama, Yoshiyuki: "Delta link homotopy for two component links, III"Journal of Mathematical Society of Japan. 55・3. 641-654 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.C.カールソン著, 金信泰造訳: "曲面・結び目・多様体のトポロジー"培風館. 200 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kanenobu, Taizo, Shima, Akiko: "Two filtrations of ribbon 2-knots"Topology Appl. Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia,1999). 121・1-2. 143-168 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kanenobu, Taizo: "Virtual arc presentations and HC-moves of ribbon 2-knots"J. Knot Theory Ramifications, Proceedings of Knots 2000 Korea, Vol.1 (Yongpyong). 11・3. 387-402 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kanenobu, Taizo: "Tangle surgeries on the double of a tangle and link polynomials"Kobe J.Math.. (出版予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Kanenobu, Taizo, Saito, Hirofusa, Satoh, Shin: "Tangles with up to seven crossings"Interdiscip.Inform.Sci.. (出版予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Kanenobu, Taizo, Nikkuni, Ryo: "Delta move and polynomial invariants of links"Topology Appl. Proceedings of the Second Joint Japan-Mexico Meeting in Topology (Matsue,2002). (出版予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Nakanishi, Yasutaka, Ohyama, Yoshiyuki: "Delta link homotopy for two component link, III"Journal of Mathematical Society of Japan. (出版予定).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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