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Discrete quantization of low-dimensional geometry with quantum invariants

Research Project

Project/Area Number 25287014
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Geometry
Research InstitutionWaseda University

Principal Investigator

Murakami Jun  早稲田大学, 理工学術院, 教授 (90157751)

Co-Investigator(Kenkyū-buntansha) 水澤 篤彦  早稲田大学, 理工学術院, 助教 (50707726)
Project Period (FY) 2013-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥8,320,000 (Direct Cost: ¥6,400,000、Indirect Cost: ¥1,920,000)
Fiscal Year 2016: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2013: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Keywords低次元トポロジー / 量子不変量 / 双曲幾何学 / 3次元多様体 / 結び目 / 体積予想 / 量子群 / 射影表現 / 結び目不変量 / 結び目理論 / 双曲幾何 / 量子展開環 / 表現論
Outline of Final Research Achievements

The aim of this research is to construct discretized quantum geometry for 2 and 3 dimensional case. To do this, various quantum invariants and their relations are studied. For example, we study the colored Jones invariant, the colored Alexander invariant, the Hennings invariant and the logarithmic invariant. Quantum invariants for knotted graphs and its relation to the geometric structure is also studied. Especially, we get the relation between quantum invariants and hyperbolic volume determined by the geometric structure for various quantum invariants.

Report

(5 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • Research Products

    (42 results)

All 2017 2016 2015 2014 2013 2012 Other

All Int'l Joint Research (6 results) Journal Article (9 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 9 results,  Acknowledgement Compliant: 4 results) Presentation (21 results) (of which Int'l Joint Research: 3 results,  Invited: 14 results) Book (1 results) Remarks (4 results) Funded Workshop (1 results)

  • [Int'l Joint Research] POSTECH/Busan National University of Education(韓国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] ETH Zurich(スイス)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Toronto University(Canada)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] IRMA, Universite de Strabourg(フランス)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] University of Toronto(カナダ)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Pohang Mathematics Institute, POSGECH(韓国)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Generalized Kashaev invariants for knots in three manifolds2017

    • Author(s)
      Murakami, Jun
    • Journal Title

      Quantum Topology

      Volume: 8 Issue: 1 Pages: 35-87

    • DOI

      10.4171/qt/86

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Yokota type invariants derived from non-integral highest weight representations of U_q(sl2)2016

    • Author(s)
      Mizusawa, Atsuhiko; Murakami, Jun
    • Journal Title

      Journal of Knot Theory and its Ramifications

      Volume: 25 Issue: 10 Pages: 1650054-1650054

    • DOI

      10.1142/s0218216516500541

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The dual Jacobian of a generalised hyperbolic tetrahedron, and volumes of prisms2016

    • Author(s)
      Kolpakov, Alexander; Murakami, Jun
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 39 Issue: 1 Pages: 45-67

    • DOI

      10.3836/tjm/1471873312

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Combinatorial Decompositions, Kirillov-Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra2016

    • Author(s)
      Kolpakov, Alexander; Murakami, Jun
    • Journal Title

      Experimental Mathematics

      Volume: online Nov 01 Issue: 2 Pages: 193-207

    • DOI

      10.1080/10586458.2016.1242441

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Invariants of handlebody-knots via Yokota's invariants2013

    • Author(s)
      Atsuhiko Mizusawa, Jun Murakami
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 22 Issue: 11 Pages: 1350068-1350068

    • DOI

      10.1142/s0218216513500685

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Optimistic limits of the colored Jones polynomials2013

    • Author(s)
      Jinseok Cho, Jun Murakami
    • Journal Title

      J. Korean Math. Soc.

      Volume: 85 Issue: 3 Pages: 641-693

    • DOI

      10.4134/jkms.2013.50.3.641

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On SL(2, C) quantum 6j-symbol and its relation to the hyperbolic volume2013

    • Author(s)
      Francesco Costantino, Jun Murakami
    • Journal Title

      Quantum Topol.

      Volume: 4 Issue: 3 Pages: 303-351

    • DOI

      10.4171/qt/41

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Linking numbers for handlebody-links2013

    • Author(s)
      Atsuhiko Mizusawa
    • Journal Title

      Proceedings of the Japan Academy, Ser. A, Mathematical Sciences

      Volume: 89 Pages: 60-62

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Volume of a doubly truncated hyperbolic tetra- hedron2012

    • Author(s)
      Kolpakov, Alexander ; Murakami,Jun
    • Journal Title

      Aequat. Math

      Volume: 電子速報版 Issue: 3 Pages: 1-15

    • DOI

      10.1007/s00010-012-0153-y

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Presentation] On the volume conjecture for quantum 6j symbols2016

    • Author(s)
      Murakami, Jun
    • Organizer
      Workshop on Teichmueler and Grothendieck-Teichmueller theories
    • Place of Presentation
      Chern Institute of Mathematics, Nankai University, Tianjin, China
    • Year and Date
      2016-07-27
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the volume conjecture of various SO(3) invariants2016

    • Author(s)
      Murakami, Jun
    • Organizer
      Topology and Analysis of Discrete Groups and Hyperbolic Spaces
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-06-24
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Volume conjecture for quantum 6j symvols2016

    • Author(s)
      Murakami, Jun
    • Organizer
      Mini workshop "Growth3"
    • Place of Presentation
      早稲田大学教育学部
    • Year and Date
      2016-04-01
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Volume formula for hyperbolic and spherical polyhedron2015

    • Author(s)
      Jun Murakami
    • Organizer
      PMI Quantum TopologySeminar
    • Place of Presentation
      Department of Mathemtics, POSTECH,Korea
    • Year and Date
      2015-12-12
    • Related Report
      2015 Annual Research Report
  • [Presentation] Volume conjecture for quantum 6j symbols2015

    • Author(s)
      Jun Murakami
    • Organizer
      PMI Quantum TopologySeminar
    • Place of Presentation
      Department of Mathemtics, POSTECH,Korea
    • Year and Date
      2015-12-11
    • Related Report
      2015 Annual Research Report
  • [Presentation] Logarithmic invariant of knots and its applications2015

    • Author(s)
      Jun Murakami
    • Organizer
      Braids, Configuration Spaces, and Quantum Topology
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2015-09-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Logarithmic invariants of knots in a three manifold2015

    • Author(s)
      Jun Murakami
    • Organizer
      New Developments in TQFT
    • Place of Presentation
      Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Denmark
    • Year and Date
      2015-07-31
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Volumes of hyperbolic an spherical polyhedrons2015

    • Author(s)
      村上 順
    • Organizer
      日本数学会2015年度年会
    • Place of Presentation
      明治大学駿河台キャンパス
    • Year and Date
      2015-03-23
    • Related Report
      2014 Annual Research Report
  • [Presentation] Volumes of hyperbolic an spherical polyhedrons2014

    • Author(s)
      Jun Murakami
    • Organizer
      8th Australia New Zealand Mathematics Convention
    • Place of Presentation
      University of Melbourne
    • Year and Date
      2014-12-08
    • Related Report
      2014 Annual Research Report
  • [Presentation] Logarithmic invariants of links2014

    • Author(s)
      Jun Murakami
    • Organizer
      Quantum Topology and Physics 2014 in Fukuoka
    • Place of Presentation
      西新プラザ,九州大学
    • Year and Date
      2014-09-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Logarithmic knot invariants and hyperbolic volumes2014

    • Author(s)
      Jun Murakami
    • Organizer
      Knots and Low Dimensional Manifolds
    • Place of Presentation
      BEXCO Convention & Exibition Center II, Busan, Korea
    • Year and Date
      2014-08-25
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Logarithmic invariant of knots2014

    • Author(s)
      Jun Murakami
    • Organizer
      Quantum Curves and Quantum Knot Invariants
    • Place of Presentation
      Banff International Research Station, Banff, Canada
    • Year and Date
      2014-06-19
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Volume conjecture for logarithmic invariant of knots2014

    • Author(s)
      Jun Murakami
    • Organizer
      Mini-workshop on the Volume conjecture
    • Place of Presentation
      KoreaInstitutefor Acvanced Study (KIAS), Seoul, Korea
    • Year and Date
      2014-06-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Generalized Kashaev invariants for knots in three-manifolds

    • Author(s)
      Jun Murakami
    • Organizer
      Modern Trends in Topological Quantum Field Theory, Workshop II
    • Place of Presentation
      Erwin Schrodinger International Institute for Mathematical Physics, Vienna, Austria
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Knots invariants coming from the small quantum group

    • Author(s)
      村上 順
    • Organizer
      第58回代数学シンポジウム
    • Place of Presentation
      広島大学理学部
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Volume formulas for a spherical tetrahedron

    • Author(s)
      Jun Murakami
    • Organizer
      Geometric structures on low-dimensional manifolds
    • Place of Presentation
      Korea Institute for Advanced Study, Seoul, 韓国
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Yokota type invariants derived from Costantino-Murakami's invariants

    • Author(s)
      水澤篤彦
    • Organizer
      Intelligence of Low-dimensional Topology
    • Place of Presentation
      京都大学 数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A homotopy classification of two-component handlebody-links

    • Author(s)
      Atsuhiko Mizusawa
    • Organizer
      International Workshop on Spatial Graphs 2013
    • Place of Presentation
      東京女子大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] Homotopy and Delta classifications of two-component handlebody-links

    • Author(s)
      Atsuhiko Mizusawa
    • Organizer
      International Conference on Topology and Geometry 2013 Joint with the 6th Japan-Mexico Topology Symposium
    • Place of Presentation
      島根大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] ハンドル体絡み目の絡み数と neighborhood homotopy

    • Author(s)
      水澤篤彦
    • Organizer
      ハンドル体結び目とその周辺 VI
    • Place of Presentation
      上智大学 四谷キャンパス
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 双曲体積と関連する空間グラフの不変量

    • Author(s)
      水澤篤彦
    • Organizer
      早稲田大学 数学若手異分野交流会 2014
    • Place of Presentation
      早稲田大学 西早稲田キャンパス
    • Related Report
      2013 Annual Research Report
  • [Book] Proceedings of Symposia in Pure Mathematics Volume 93, String-Math 20142016

    • Author(s)
      VIncent Bouchard et al
    • Total Pages
      418
    • Publisher
      American Mathematical Society
    • Related Report
      2016 Annual Research Report
  • [Remarks] 研究について(村上 順)

    • URL

      http://www.f.waseda.jp/murakami/research-jp.html

    • Related Report
      2016 Annual Research Report
  • [Remarks] 研究について

    • URL

      http://www.f.waseda.jp/murakami/research-jp.html

    • Related Report
      2015 Annual Research Report
  • [Remarks] Web Page of Jun MURAKAMI

    • URL

      http://www.f.waseda.jp/murakami/jun-home-j.html

    • Related Report
      2014 Annual Research Report
  • [Remarks] 研究について(村上 順)

    • URL

      http://www.f.waseda.jp/murakami/research-jp.html

    • Related Report
      2013 Annual Research Report
  • [Funded Workshop] Workshop on Volume Conjecture and Quantum Topology2016

    • Place of Presentation
      早稲田大学西早稲田キャンパス
    • Year and Date
      2016-09-06
    • Related Report
      2016 Annual Research Report

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Published: 2013-05-21   Modified: 2022-02-16  

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