Quantum invariants of knots and representations of knot groups

• Project/Area Number: 26400079
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku 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: ¥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.

Research Products

• [Int'l Joint Research] テキサス大学ダラス校(米国)
• [Int'l Joint Research] Universite Denis Diderot - Paris 7(フランス)
• [Journal Article] Root polytopes, parking functions, and the HOMFLY polynomial (2017)
  • Author(s): T. Kalman and H. Murakami
  • Journal Title: Quantum Topology Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Hecke-type formulas for families of unified Witten-Reshetikhin-Turaev invariants (2017)
  • Author(s): K.Hikami and J.Lovejoy
  • Journal Title: Communications in Number Theory and Physics Volume: 11
  • 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 recursion (2017)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Partial chord diagrams and matrix models (2017)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Enumeration of chord diagrams via topo logical recursion and quantum curve techniques (2017)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Braids, complex volume and cluster algebras (2015)
  • 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
  • Peer Reviewed
• [Journal Article] Torus knots and quantum modular forms (2015)
  • 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
  • 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 knot (2015)
  • Author(s): H. Murakami
  • Journal Title: Acta Math. Vietnam Volume: 39 Issue: 4 Pages: 649-710
  • DOI: 10.1007/s40306-014-0084-x
  • Peer Reviewed
• [Journal Article] Braids, complex volume, and cluster algebra (2015)
  • Author(s): Kazuhiro Hikami and Rei Inoue
  • Journal Title: Algebraic and Geometric Topology Volume: -
  • Peer Reviewed
• [Journal Article] Braiding operator via quantum cluster algebra (2014)
  • 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
  • Peer Reviewed / Open Access
• [Presentation] Fatgraph models for RNA molecules (2017)
  • Author(s): H. Fuji
  • Organizer: iTHEMS seminar
  • Place of Presentation: 理化学研究所（和光）
  • Year and Date: 2017-03-23
  • Int'l Joint Research / Invited
• [Presentation] Partial chord diagrams and matrix models (2017)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] q-series and quantum invariants (2017)
  • Author(s): K.Hikami
  • Organizer: Frontiers in Mathematical Physics
  • Place of Presentation: 立教大学（東京）
  • Year and Date: 2017-01-06
  • Int'l Joint Research / Invited
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion (2016)
  • 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
  • Invited
• [Presentation] The colored Jones polynomial and representations of a knot group (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Colored Jones polynomial and SL(2;C) representations of a knot group (2016)
  • Author(s): H. Murakami
  • Organizer: Knots in Hellas 2016
  • Place of Presentation: International Olympic Academy, Ancient Olympia, Greece
  • Year and Date: 2016-07-21
  • Int'l Joint Research / Invited
• [Presentation] On a generalization of the volume conjecture for cable knots (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Quantum modular form from knots and 3-manifolds (2016)
  • Author(s): K.Hikami
  • Organizer: Number Theory and Physics
  • Place of Presentation: Institut Henri Poincare, France
  • Year and Date: 2016-05-23
  • Int'l Joint Research / Invited
• [Presentation] Quantu modular form (2016)
  • Author(s): K. Hikami
  • Organizer: Number Theory Seminar
  • Place of Presentation: Pohang University of Science and Technology, Korea
  • Year and Date: 2016-03-16
  • Invited
• [Presentation] Quantum dilogarithm and quantum invariant (2016)
  • Author(s): K. Hikami
  • Organizer: Quantum Topology Seminar
  • Place of Presentation: Pohang University of Science and Technology, Korea
  • Year and Date: 2016-03-15
  • Invited
• [Presentation] Quantum modular forms and quantum invariants (2015)
  • 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
  • Invited
• [Presentation] Enumerating chord diagrams via matrix models (2015)
  • 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
  • Invited
• [Presentation] On the enumeration of chord diagrams and the BPZ equation via the matrix model (2015)
  • Author(s): H. Fuji
  • Organizer: Quantization of Spectral Curves
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2015-11-02
  • Invited
• [Presentation] Enumerating chord diagrams via matrix models (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Matrix Model Approach to RNA (2015)
  • 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
  • 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
  • Invited
• [Presentation] The colored Jones polynomials, the Chern--Simons invariants, and the Reidemeister torsions of a knot (2015)
  • Author(s): H. Murakami
  • Organizer: Invariants in Low Dimensional Geometry
  • Place of Presentation: Gazi University, Ankara, Turkey
  • Year and Date: 2015-08-11
  • Int'l Joint Research / Invited
• [Presentation] The volume conjecture for non-simple knots (2015)
  • Author(s): H. Murakami
  • Organizer: First Encounter to Quantum Topology
  • Place of Presentation: Korea Institute for Advanced Study
  • Year and Date: 2015-07-20
  • 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
  • Invited
• [Presentation] Quantum modular form from torus knots (2014)
  • 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
  • Invited
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a cable knot (2014)
  • 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
  • Invited
• [Presentation] On the volume conjecture and its generalizations (2014)
  • 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
  • Invited