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Study of perturbative quantum invariant of rational homology 3-spheres

Research Project

Project/Area Number 22540073
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKyushu University

Principal Investigator

TAKATA Toshie  九州大学, 数理学研究院, 准教授 (40253398)

Project Period (FY) 2010 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords低次元トポロジー / 量子不変量 / 3次元多様体 / free energy / 3次元多様体 / LMO不変量 / weight system
Research Abstract

For a rational homology 3-sphre, the SU(N) free energy is a power series in 2 variables and an topological invariant of M. The SU(N) free energy can be obtained by applying the SU(N) weight system to the LMO invariant of M. By using a combinatorial way to compute the SU(N) weight system, we obtained an formula for any genus of the SU(N) free energy for Seifert homology 3-spheres. It can be written in terms of the generalized Catalan numbers. Especially, the genus 0 part of the free energy can be written in terms of the generalized Catalan numbers for genus 0 where parameters are even numbers. We gave a formula of these generalized Catalan numbers.

Report

(4 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (10 results)

All 2012 2011 2010

All Journal Article (2 results) (of which Peer Reviewed: 2 results) Presentation (8 results)

  • [Journal Article] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere2011

    • Author(s)
      高田敏恵
    • Journal Title

      International Journal of Mathematics

      Volume: 22 Pages: 465-482

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere2011

    • Author(s)
      高田敏恵
    • Journal Title

      International Joounal of Mathematics

      Volume: 22 Issue: 04 Pages: 465-482

    • DOI

      10.1142/s0129167x11006866

    • Related Report
      2011 Annual Research Report 2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] An algebraic formula of the Witten-Reshetikihin-Turaev invariant for 3-manifolds2012

    • Author(s)
      高田敏恵
    • Organizer
      The 8th East Asian School of Knots and Related Topics
    • Place of Presentation
      KAIST(韓国)
    • Year and Date
      2012-01-09
    • Related Report
      2012 Final Research Report
  • [Presentation] An algebraic formula of the Witten-Reshetikihin-Turaev invariant for 3-manifods2012

    • Author(s)
      高田敏恵
    • Organizer
      The 8the East Asian School of Knots and Related Topics
    • Place of Presentation
      KAIST(韓国)
    • Year and Date
      2012-01-09
    • Related Report
      2011 Annual Research Report
  • [Presentation] Ribbon Hopf algebras と3次元多様体の量子不変量2011

    • Author(s)
      高田敏恵
    • 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
      2011-08-24
    • Related Report
      2012 Final Research Report
  • [Presentation] Ribbon Hopf algebrasと3次元多様体の量子不変量2011

    • Author(s)
      高田敏恵
    • 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(東京)
    • Year and Date
      2011-08-24
    • Related Report
      2011 Annual Research Report
  • [Presentation] 3 次元多様体の LMO 不変量と摂動的不変量について2010

    • Author(s)
      高田敏恵
    • Organizer
      日本数学会 2010 年度秋季総合分科会トポロジー分科会特別講演
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-22
    • Related Report
      2012 Final Research Report
  • [Presentation] 3次元多様体のLMO不変量と摂動的不変量について2010

    • Author(s)
      高田敏恵
    • Organizer
      日本数学会2010年度秋季総合分科会トポロジー分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-22
    • Related Report
      2010 Annual Research Report
  • [Presentation] A relation between the HOMFLY and Kauffman polynomials for 2-component torus links2010

    • Author(s)
      高田敏恵
    • Organizer
      広島大学トポロジー・幾何セミナー
    • Place of Presentation
      広島大学
    • Year and Date
      2010-07-13
    • Related Report
      2012 Final Research Report
  • [Presentation] On the SO(N) and Sp(N) free energy of a rational homology 3-sphere2010

    • Author(s)
      高田敏恵
    • Organizer
      量子群と量子トポロジー
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-04-20
    • Related Report
      2012 Final Research Report 2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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