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Handlebody-knots and topology

Research Project

Project/Area Number 21740035
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

ISHII Atsushi  筑波大学, 数理物質系, 助教 (00531451)

Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords位相幾何学 / 結び目理論
Research Abstract

A handlebody-knot is a handlebody embedded in the 3-sphere. Two handlebody-knots are equivalent if one can be transformed into the other by an isotopy of the 3-sphere. A handlebody-knot is a natural generalization of a knot, and is closely related to a spatial graph and a 3-manifold. The results of this study are as follows : I defined a normalized Yamada polynomial. I provided methods to detect the irreducibility of a handlebody-knot by using quandle coloring invariants. I introduced the IH-complex on the set of spatial trivalent graphs, and evaluated the IH-distance by using invariants.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (24 results)

All 2012 2011 2010 2009

All Journal Article (9 results) (of which Peer Reviewed: 9 results) Presentation (15 results)

  • [Journal Article] On normalizations of a regular isotopy invariant for spatial graphs2011

    • Author(s)
      Atsushi Ishii
    • Journal Title

      Internat. J. Math

      Volume: 22 Issue: 11 Pages: 1545-1559

    • DOI

      10.1142/s0129167x1100729x

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The quandle coloring invariant of a reducible handlebody-knot2011

    • Author(s)
      Atsushi Ishii, Kengo Kishimoto
    • Journal Title

      Tsukuba J. Math

      Volume: 35 Pages: 131-141

    • NAID

      120006312315

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The quandle coloring invariant of a reducible handlebody-knot2011

    • Author(s)
      Atsushi Ishii, Kengo Kishimoto
    • Journal Title

      Tsukuba J.Math.

      Volume: 35 Pages: 131-141

    • NAID

      120006312315

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The IH-complex of spatial trivalent graphs2010

    • Author(s)
      Atsushi Ishii, Kengo Kishimoto
    • Journal Title

      Tokyo J. Math

      Volume: 33 Issue: 2 Pages: 523-535

    • DOI

      10.3836/tjm/1296483486

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The leading finite type coefficients of the Links-Gould polynomial of a link2010

    • Author(s)
      Atsushi Ishii
    • Journal Title

      Kyungpook Math. J

      Volume: 50 Pages: 49-58

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Miyazawa polynomial for long virtual knots2010

    • Author(s)
      Atsushi Ishii, Naoko Kamada, Seiichi Kamada
    • Journal Title

      Topology Appl

      Volume: 157 Issue: 1 Pages: 290-297

    • DOI

      10.1016/j.topol.2009.04.058

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The IH-complex of spatial trivalent graphs2010

    • Author(s)
      Atsushi Ishii, Kengo Kishimoto
    • Journal Title

      Tokyo J.Math.

      Volume: 33 Pages: 523-535

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The leading finite type coefficients of the Links-Gould polynomial of a link2010

    • Author(s)
      Atsushi Ishii
    • Journal Title

      Kyungpook Math.J.

      Volume: 50 Pages: 49-58

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Miyazawa polynomial for long virtual knots2010

    • Author(s)
      Atsushi Ishii, Naoko Kamada, Seiichi Kamada
    • Journal Title

      Topology Appl. 157

      Pages: 290-297

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Quantum-commutative quantum-symmetric algebras and invariants for handlebody-links2012

    • Author(s)
      Atsushi Ishii
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-26
    • Related Report
      2011 Final Research Report
  • [Presentation] An introduction to handlebody-knot theory, Colloquium2012

    • Author(s)
      Atsushi Ishii
    • Place of Presentation
      The University of South Florida, The United States of America
    • Year and Date
      2012-02-13
    • Related Report
      2011 Final Research Report
  • [Presentation] Invariants for handlebody-knots derived from Yetter-Drinfeld modules2012

    • Author(s)
      Atsushi Ishii
    • Organizer
      The 8th East Asian School of Knots and Related Topics, Korea Advanced Institute of Science and Technology
    • Place of Presentation
      Daejeon, Korea
    • Year and Date
      2012-01-11
    • Related Report
      2011 Final Research Report
  • [Presentation] A handlebody-knot and a Yetter-Drinfeld module2011

    • Author(s)
      Atsushi Ishii
    • Organizer
      結び目の量子不変量とそのカテゴリー化
    • Place of Presentation
      早稲田大学
    • Year and Date
      2011-08-26
    • Related Report
      2011 Final Research Report
  • [Presentation] A quandle cocycle invariant with non-commutative flows for a handlebody-knot2011

    • Author(s)
      Atsushi Ishii
    • Organizer
      Intelligence of Low-dimensional Topology
    • Place of Presentation
      京都大学
    • Year and Date
      2011-05-27
    • Related Report
      2011 Final Research Report
  • [Presentation] A quandle cocycle invariant with non-commutative flows for a handle body-knot2011

    • Author(s)
      石井敦
    • Organizer
      Intelligence of Low-dimensional Topology
    • Place of Presentation
      京都大学(京都市)(招待講演)
    • Year and Date
      2011-05-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] Quandle colorings with non-commutative flows2011

    • Author(s)
      Atsushi Ishii
    • Organizer
      トポロジー火曜セミナー
    • Place of Presentation
      東京大学
    • Year and Date
      2011-05-17
    • Related Report
      2011 Final Research Report
  • [Presentation] 非可換フローを用いたハンドル体結び目のカンドルコサイクル不変量2011

    • Author(s)
      Atsushi Ishii
    • Organizer
      Knotting Nagoya
    • Place of Presentation
      名古屋市立大学
    • Year and Date
      2011-02-19
    • Related Report
      2011 Final Research Report
  • [Presentation] A quandle cocycle invariant on a group representation for handlebody-links I, II2011

    • Author(s)
      Atsushi Ishii, Yeonhee Jang, Kanako Oshiro
    • Organizer
      The Seventh East Asian School of Knots and Related Topics
    • Place of Presentation
      広島大学
    • Year and Date
      2011-01-10
    • Related Report
      2011 Final Research Report
  • [Presentation] A knotted handlebody and a spatial graph2010

    • Author(s)
      Atsushi Ishii
    • Organizer
      International Workshop on Spatial Graphs 2010
    • Place of Presentation
      早稲田大学
    • Year and Date
      2010-08-19
    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Presentation] ハンドル体絡み目の有限型不変量2010

    • Author(s)
      Atsushi Ishii
    • Organizer
      ハンドル体結び目とその周辺III
    • Place of Presentation
      東京女子大学
    • Year and Date
      2010-05-15
    • Related Report
      2011 Final Research Report
  • [Presentation] Invariants for knotted handlebodies2010

    • Author(s)
      Atsushi Ishii
    • Organizer
      量子群と量子トポロジー
    • Place of Presentation
      京都大学
    • Year and Date
      2010-04-20
    • Related Report
      2011 Final Research Report
  • [Presentation] On normalizations of a regular isotopy invariant for spatial graphs2010

    • Author(s)
      Atsushi Ishii
    • Organizer
      日本数学会年会
    • Place of Presentation
      慶應義塾大学
    • Year and Date
      2010-03-26
    • Related Report
      2011 Final Research Report
  • [Presentation] Normalizations for the Yamada polynomial2010

    • Author(s)
      Atsushi Ishii
    • Organizer
      The Sixth East Asian School of Knots and Related Topics
    • Place of Presentation
      Nankai University, Tianjin, China
    • Year and Date
      2010-01-26
    • Related Report
      2011 Final Research Report
  • [Presentation] The IH-complex of spatial trivalent graphs I2009

    • Author(s)
      石井敦
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-25
    • Related Report
      2009 Annual Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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