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Quantum invariants of knots and representations of knot groups

Research Project

Project/Area Number 26400079
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

Murakami Hitoshi  東北大学, 情報科学研究科, 教授 (70192771)

Co-Investigator(Kenkyū-buntansha) 樋上 和弘  九州大学, 数理学研究院, 准教授 (60262151)
藤 博之  香川大学, 教育学部, 准教授 (50391719)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords体積予想 / Jons多項式 / 結び目 / Chern-Simons不変量 / Reidemeister torsion / 量子位相幾何学 / 低次元位相幾何学 / 色付きJones多項式 / torus 結び目 / Chern-Simons 不変量 / 基本群の表現 / トーラス結び目 / ジョーンズ多項式 / チャーン・サイモンズ不変量 / ライデマイスターの捩れ
Outline of Final Research Achievements

In this research I studied mainly on the volume conjecture and its generalization about the colored Jones polynomial of a knot.
As a result, I proved a generalization of the volume conjecture, that is, I proved that the asymptotic behavior of the colored Jones polynomial gives the Chern-Simons invariant and the Reidemeister torsion of the knot complement.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (34 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (2 results) Journal Article (10 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 10 results,  Acknowledgement Compliant: 6 results,  Open Access: 6 results) Presentation (22 results) (of which Int'l Joint Research: 10 results,  Invited: 22 results)

  • [Int'l Joint Research] テキサス大学ダラス校(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Universite Denis Diderot - Paris 7(フランス)

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Root polytopes, parking functions, and the HOMFLY polynomial2017

    • Author(s)
      T. Kalman and H. Murakami
    • Journal Title

      Quantum Topology

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Hecke-type formulas for families of unified Witten-Reshetikhin-Turaev invariants2017

    • Author(s)
      K.Hikami and J.Lovejoy
    • Journal Title

      Communications in Number Theory and Physics

      Volume: 11

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] The boundary length and point spectrum enumeration of pa rtial chord diagrams using cut and join recursion2017

    • Author(s)
      J. E. Andersen, H. Fuji, R. C. Penner, and C. M. Reidys
    • Journal Title

      Travaux Mathematiques, University of Luxembourg

      Volume: 25 Pages: 213-232

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Partial chord diagrams and matrix models2017

    • Author(s)
      J. E. Andersen, H. Fuji, M. Manabe, R. C . Penner, and P. Sulkowski
    • Journal Title

      Travaux Mathematiques, University of Luxembourg

      Volume: 25 Pages: 233-283

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Enumeration of chord diagrams via topo logical recursion and quantum curve techniques2017

    • Author(s)
      J. E. Andersen, H. Fuji, M. Manabe, R. C . Penner, and P. Sulkowski
    • Journal Title

      Travaux Mathematiques, University of Luxembourg

      Volume: 25 Pages: 285-323

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Braids, complex volume and cluster algebras2015

    • Author(s)
      K. Hikami and R. Inoue
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 15 Issue: 4 Pages: 2175-2194

    • DOI

      10.2140/agt.2015.15.2175

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Torus knots and quantum modular forms2015

    • Author(s)
      K. Hikami and J. Lovejoy
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 2 Issue: 1 Pages: 1-15

    • DOI

      10.1186/s40687-014-0016-3

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a twice-iterated torus knot2015

    • Author(s)
      H. Murakami
    • Journal Title

      Acta Math. Vietnam

      Volume: 39 Issue: 4 Pages: 649-710

    • DOI

      10.1007/s40306-014-0084-x

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Braids, complex volume, and cluster algebra2015

    • Author(s)
      Kazuhiro Hikami and Rei Inoue
    • Journal Title

      Algebraic and Geometric Topology

      Volume: -

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Braiding operator via quantum cluster algebra2014

    • Author(s)
      Kazuhiro Hikami, Rei Inoue
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 47 Issue: 47 Pages: 1-21

    • DOI

      10.1088/1751-8113/47/47/474006

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Fatgraph models for RNA molecules2017

    • Author(s)
      H. Fuji
    • Organizer
      iTHEMS seminar
    • Place of Presentation
      理化学研究所(和光)
    • Year and Date
      2017-03-23
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Partial chord diagrams and matrix models2017

    • Author(s)
      H. Fuji
    • Organizer
      Colloquium“The Algebra and Geometry of Modern Physics”
    • Place of Presentation
      University of Warsaw, Poland
    • Year and Date
      2017-03-09
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] q-series and quantum invariants2017

    • Author(s)
      K.Hikami
    • Organizer
      Frontiers in Mathematical Physics
    • Place of Presentation
      立教大学(東京)
    • Year and Date
      2017-01-06
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion2016

    • Author(s)
      H. Murakami
    • Organizer
      東北結び目セミナー2016
    • Place of Presentation
      東北大学(仙台)
    • Year and Date
      2016-10-15
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] The colored Jones polynomial and representations of a knot group2016

    • Author(s)
      H. Murakami
    • Organizer
      Workshop on Volume Conjecture and Quantum Topology
    • Place of Presentation
      早稲田大学(東京)
    • Year and Date
      2016-09-08
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Colored Jones polynomial and SL(2;C) representations of a knot group2016

    • Author(s)
      H. Murakami
    • Organizer
      Knots in Hellas 2016
    • Place of Presentation
      International Olympic Academy, Ancient Olympia, Greece
    • Year and Date
      2016-07-21
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On a generalization of the volume conjecture for cable knots2016

    • Author(s)
      H. Murakami
    • Organizer
      2016 International Coference on Low-Dimensional Topology
    • Place of Presentation
      Southwest Jiaotong University, Emei, Sichuan, China
    • Year and Date
      2016-06-13
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantum modular form from knots and 3-manifolds2016

    • Author(s)
      K.Hikami
    • Organizer
      Number Theory and Physics
    • Place of Presentation
      Institut Henri Poincare, France
    • Year and Date
      2016-05-23
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantu modular form2016

    • Author(s)
      K. Hikami
    • Organizer
      Number Theory Seminar
    • Place of Presentation
      Pohang University of Science and Technology, Korea
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Quantum dilogarithm and quantum invariant2016

    • Author(s)
      K. Hikami
    • Organizer
      Quantum Topology Seminar
    • Place of Presentation
      Pohang University of Science and Technology, Korea
    • Year and Date
      2016-03-15
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Quantum modular forms and quantum invariants2015

    • Author(s)
      K. Hikami
    • Organizer
      九州大学代数セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2015-12-11
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Enumerating chord diagrams via matrix models2015

    • Author(s)
      H. Fuji
    • Organizer
      RMT2015: Random matrix theory from fundamental mathematics to biological applications
    • Place of Presentation
      沖縄科学技術大学院大学
    • Year and Date
      2015-11-06
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On the enumeration of chord diagrams and the BPZ equation via the matrix model2015

    • Author(s)
      H. Fuji
    • Organizer
      Quantization of Spectral Curves
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2015-11-02
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Enumerating chord diagrams via matrix models2015

    • Author(s)
      H. Fuji
    • Organizer
      :96e rencontre entre mathematiciens et physiciens theoriciens : Geometrie et biophysique
    • Place of Presentation
      IRMA, ストラスブール大学
    • Year and Date
      2015-09-17
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Matrix Model Approach to RNA2015

    • Author(s)
      H. Fuji
    • Organizer
      離散的手法による場と時空のダイナミクス
    • Place of Presentation
      岡山衛生会館
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On the volume conjecture and its generalizations'2015

    • Author(s)
      H. Murakami
    • Organizer
      拡大KOOKセミナー2015
    • Place of Presentation
      神戸大学
    • Year and Date
      2015-08-21
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] The colored Jones polynomials, the Chern--Simons invariants, and the Reidemeister torsions of a knot2015

    • Author(s)
      H. Murakami
    • Organizer
      Invariants in Low Dimensional Geometry
    • Place of Presentation
      Gazi University, Ankara, Turkey
    • Year and Date
      2015-08-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The volume conjecture for non-simple knots2015

    • Author(s)
      H. Murakami
    • Organizer
      First Encounter to Quantum Topology
    • Place of Presentation
      Korea Institute for Advanced Study
    • Year and Date
      2015-07-20
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] モックテータ函数の周辺2015

    • Author(s)
      樋上 和弘
    • Organizer
      東北大学情報数理談話会
    • Place of Presentation
      東北大学大学院情報科学研究科
    • Year and Date
      2015-01-16
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Quantum modular form from torus knots2014

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Low-dimensional Topology and Number Theory
    • Place of Presentation
      The Mathematisches Forschungsinstitut Oberwolfach, Germany
    • Year and Date
      2014-08-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a cable knot2014

    • Author(s)
      Hitoshi Murakami
    • Organizer
      Low-dimensional Topology and Number Theory
    • Place of Presentation
      The Mathematisches Forschungsinstitut Oberwolfach, Germany
    • Year and Date
      2014-08-20
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the volume conjecture and its generalizations2014

    • Author(s)
      Hitoshi Murakami
    • Organizer
      The 2014 KMJ Conference for Accreditation Strategies
    • Place of Presentation
      Kyungpook National University, Daegu, Korea
    • Year and Date
      2014-08-08
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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