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Quantum topology and invariants of knots and 3-manifolds related to gauge theory

Research Project

Project/Area Number 16K13754
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionKyoto University

Principal Investigator

Ohtsuki Tomotada  京都大学, 数理解析研究所, 教授 (50223871)

Co-Investigator(Kenkyū-buntansha) 佐伯 修  九州大学, マス・フォア・インダストリ研究所, 教授 (30201510)
Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords結び目 / 3次元多様体 / 不変量 / 低次元トポロジー / 量子トポロジー
Outline of Final Research Achievements

The author calculated the asymptotic expansion of the Kashaev invariant for hyperbolic knots with up to 7 crossings, and proved the volume conjecture for these knots. Further, the author proved the volume conjecture for hyperbolic 3-manifolds obtained by integral surgery along the figure-eight knot. Furthermore, the author obsearved that the Reidemeister torsion appears in the semi-classical part of the asymptotic expansion of the quantum invariant, and proved that for some 3-manifolds.
The author held the workshop "Intelligence of Low-dimensional Topology" at RIMS in each year.

Academic Significance and Societal Importance of the Research Achievements

1970年代にはじまった双曲幾何の研究と1980年代にはじまった量子トポロジーの研究は、それぞれ別々に発展してきたが、1999年に発見された体積予想はそれらを結びつける重要な予想である。体積予想は、数理物理的には、Chern-Simons 経路積分に対して無限次元の鞍点法を形式的に適用することによって導かれ、その摂動展開には双曲体積や Reidemeister torsion が現われる、筆者は、いくつかの結び目と3次元多様体に対して、体積予想を証明した。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (41 results)

All 2020 2019 2018 2017 2016

All Journal Article (20 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 18 results,  Open Access: 4 results,  Acknowledgement Compliant: 2 results) Presentation (20 results) (of which Int'l Joint Research: 13 results,  Invited: 18 results) Funded Workshop (1 results)

  • [Journal Article] Linking between singular locus and regular fibers2020

    • Author(s)
      Osamu Saeki
    • Journal Title

      Journal of Singularities

      Volume: 21 Pages: 234-248

    • DOI

      10.5427/jsing.2020.21n

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the quantum SU(2) invariant at q=exp(4πi/N) and the twisted Reidemeister torsion for some closed 3-manifolds2019

    • Author(s)
      大槻知忠,高田敏恵
    • Journal Title

      Communications in Mathematical Physics

      Volume: 370 Issue: 1 Pages: 151-204

    • DOI

      10.1007/s00220-019-03489-2

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Elimination of definite fold II2019

    • Author(s)
      Saeki Osamu
    • Journal Title

      Kyushu J. Math.

      Volume: 73 Pages: 239-250

    • NAID

      130007871384

    • Related Report
      2019 Annual Research Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] A signature invariant for stable maps of 3-manifolds into surfaces2019

    • Author(s)
      Saeki Osamu
    • Journal Title

      Rev. Roumaine Math. Pures Appl.

      Volume: 64 Pages: 541-563

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A signature invariant for stable maps of 3-manifolds into surfaces2019

    • Author(s)
      Saeki Osamu
    • Journal Title

      Proceedings of the Australian-Japanese Workshop on Real and Complex Singularities

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the asymptotic expansion of the quantumSU(2) invariant at q =exp(4π?N) for closed hyperbolic 3?manifolds obtained byintegral surgery along the figure-eight knot2018

    • Author(s)
      Ohtsuki Tomotada
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 18 Issue: 7 Pages: 4187-4274

    • DOI

      10.2140/agt.2018.18.4187

    • Related Report
      2019 Annual Research Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Simplified broken Lefschetz fibrations and trisections of 4-manifolds2018

    • Author(s)
      Baykur R. ?nan?、Saeki Osamu
    • Journal Title

      Proceedings of the National Academy of Sciences

      Volume: 115 Issue: 43 Pages: 10894-10900

    • DOI

      10.1073/pnas.1717175115

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings2018

    • Author(s)
      T. Ohtsuki, Y. Yokota
    • Journal Title

      Math. Proc. Camb. Phil. Soc.

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Simplified broken Lefschetz fibrations and trisections of 4-manifolds2018

    • Author(s)
      R.I. Baykur and O. Saeki
    • Journal Title

      Proc. Natl. Acad. Sci. USA.

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Singular fibers of stable maps of manifold pairs and their applications2018

    • Author(s)
      O. Saeki and T. Yamamoto
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 222 Pages: 259-294

    • DOI

      10.1007/978-3-319-73639-6_8

    • ISBN
      9783319736389, 9783319736396
    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings2017

    • Author(s)
      OHTSUKI TOMOTADA、YOKOTA YOSHIYUKI
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: - Issue: 2 Pages: 1-53

    • DOI

      10.1017/s0305004117000494

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the On the asymptotic expansion of the Kashaev invariant of the hyperbolic knots with seven crossings2017

    • Author(s)
      T. Ohtsuki
    • Journal Title

      Internat. J. Math.

      Volume: 28 Issue: 13 Pages: 1750096-1750096

    • DOI

      10.1142/s0129167x17500963

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Theory of singular fibers and Reeb spaces for visualization2017

    • Author(s)
      O. Saeki
    • Journal Title

      Topological Methods in Data Analysis and Visualization IV - Theory, Algorithms, and Applications

      Volume: 報告集 Pages: 3-33

    • DOI

      10.1007/978-3-319-44684-4_1

    • ISBN
      9783319446820, 9783319446844
    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Vassiliev type invariant of order one for stable maps of 3-manifolds into surfaces2017

    • Author(s)
      佐伯修
    • Journal Title

      数理解析研究所講究録

      Volume: 2049 Pages: 155-172

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] On the asymptotic expansion of the Kashaev invariant of the 5_2 knot2016

    • Author(s)
      T. Ohtsuki
    • Journal Title

      Quantum Topology

      Volume: 7 Issue: 4 Pages: 669-735

    • DOI

      10.4171/qt/83

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Problems on Low-dimensional Topology, 20162016

    • Author(s)
      T. Ohtsuki (ed)
    • Journal Title

      RIMS Kokyuroku

      Volume: 2004 Pages: 115-129

    • Related Report
      2016 Research-status Report
    • Open Access / Acknowledgement Compliant
  • [Journal Article] Topology of manifolds and global theory of singularities2016

    • Author(s)
      O. Saeki
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B55 Pages: 185-203

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Singular fibers of stable maps of 3-manifolds with boundary into surfaces and their applications2016

    • Author(s)
      Osamu Saeki and Takahiro Yamamoto
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 16 Issue: 3 Pages: 1379-1402

    • DOI

      10.2140/agt.2016.16.1379

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Cobordism group of Morse functions on surfaces with boundary2016

    • Author(s)
      O. Saeki and T. Yamamoto
    • Journal Title

      Contemporary Mathematics

      Volume: 675 Pages: 279-297

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multivariate topology simplification2016

    • Author(s)
      A. Chattopadhyay, H. Carr, D. Duke, Z. Geng, and Osamu Saeki
    • Journal Title

      Computational Geometry

      Volume: 58 Pages: 1-24

    • DOI

      10.1016/j.comgeo.2016.05.006

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Reeb graphs of smooth functions on manifolds2019

    • Author(s)
      O. Saeki
    • Organizer
      可微分写像の特異点論とその応用
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Data visualization using differential topology2019

    • Author(s)
      O. Saeki
    • Organizer
      International Joint Conference on AI & Data Science
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Manifolds admitting fold-cusp maps of certain restricted indices2019

    • Author(s)
      O. Saeki
    • Organizer
      特異点論とトポロジー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Unlinking singular loci from regular fibers and its application to submersions2019

    • Author(s)
      O. Saeki
    • Organizer
      Lefschetz Pencils and Low dimensional Topology
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Data visualization using differential topology2019

    • Author(s)
      O. Saeki
    • Organizer
      National Taiwan Normal University(NTNU)-Kyushu University Joint Forum
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Simplified broken Lefschetz fibrations and trisections of 4-manifolds2018

    • Author(s)
      O. Saeki
    • Organizer
      Intelligence of Low-dimensional Topology
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Singular locus and regular fibers, do they link each other ?2018

    • Author(s)
      O. Saeki
    • Organizer
      15th International Workshop on Real and Complex Singularities
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Elimination of definite fold for simple stable maps2018

    • Author(s)
      O. Saeki
    • Organizer
      Real Algebraic Geometry and Singularity Theory Symposium
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Unlinking singular locus from regular fibers and its application to submersions2018

    • Author(s)
      O. Saeki
    • Organizer
      Geometry, Topology and Dynamics Seminar
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Simplifying indefinite fibrations and trisections of 4-manifolds2018

    • Author(s)
      O. Saeki
    • Organizer
      The 13th Kagoshima Algebra-Analysis-Geometry Seminar
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Introduction to singularity theory and fiber topology in multivariate data analysis2017

    • Author(s)
      O. Saeki
    • Organizer
      Topology, Computation and Data Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Indefinite fibrations on differentiable 4-manifolds2017

    • Author(s)
      O. Saeki
    • Organizer
      Brazil-Mexico 3rd Meeting on Singularities
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A Vassiliev type invariant of order one for stable maps of 3-manifolds into surfaces2017

    • Author(s)
      O. Saeki
    • Organizer
      PRIMA 3rd Congress, Singularities of Spaces and Mappings
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Topologia das singularidades e teoria de n\'{o}s2017

    • Author(s)
      O. Saeki
    • Organizer
      IV Encontro de Singularidades no Nordeste
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 安定写像の特異ファイバーとその応用2017

    • Author(s)
      佐伯修
    • Organizer
      RIMS-IMI談話会
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On the asymptotic expansion of the quantum SU(2) invariant at q = \exp(4\pi/N) for closed hyperbolic 3-manifolds obtained by integral surgery along the figure-eight knot2016

    • Author(s)
      T. Ohtsuki
    • Organizer
      Volume conjecture and quantum topology
    • Place of Presentation
      早稲田大学(東京都新宿区)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 可微分多様体上の安定写像のトポロジー2016

    • Author(s)
      佐伯修
    • Organizer
      特異点の大域的研究
    • Place of Presentation
      兵庫教育大学(兵庫県神戸市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Stable maps on 3-manifolds and signatures of 4-manifolds with boundary2016

    • Author(s)
      佐伯修
    • Organizer
      4次元トポロジー
    • Place of Presentation
      大阪市立大学(大阪府大阪市)
    • Related Report
      2016 Research-status Report
  • [Presentation] A Vassiliev type invariant of order one for stable maps of 3-manifolds into surfaces2016

    • Author(s)
      佐伯修
    • Organizer
      可微分写像の特異点論とその応用
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Indefinite fibrations on 4-manifolds2016

    • Author(s)
      佐伯修
    • Organizer
      Differential Geometry, Lie Theory and Low Dimensional Topology
    • Place of Presentation
      La Trobe University (Melbourne, Australia)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] East Asian Conference on Geometric Topology2020

    • Related Report
      2019 Annual Research Report

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Published: 2016-04-21   Modified: 2023-03-16  

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