Volume Conjecture and its generalizations

• Project/Area Number: 22540069
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: MURAKAMI Hitoshi 東京工業大学, 大学院・理工学研究科, 准教授 (70192771)
• Co-Investigator(Kenkyū-buntansha): HIKAMI Kazuhiro 九州大学, 数理(科)学研究科(研究院), 准教授 (60262151)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 位相幾何 / 結び目 / 体積予想 / Jones多項式 / 色付きJones多項式 / 体積 / Chern-Simons不変量 / Reidemeister torsion / colored Jones 多項式 / HOMFLY多項式 / Chem-Simons不変量 / 基本群の表現

Research Abstract

The volume conjecture states that by studying a certain asymptotic behavior of the colored Jones polynomial of a knot would tell us the volume of the knot complement. It was proposed by R. Kashaev, J. Murakami and the main investigator. Now the conjecture is generalized to various ways and attracts many researchers including theoretical physicists. In this research we solve part of the conjecture, give yet more generalizations, and solve part of these generalizations.

Research Products

• [Journal Article] The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot (2013)
  • Author(s): Hitoshi Murakami
  • Journal Title: J.Topology Volume: 6 Issue: 1 Pages: 193-216
  • DOI: 10.1112/jtopol/jts036
  • Peer Reviewed
• [Journal Article] Twisted elliptic genus for K3 and Borcherds product (2012)
  • Author(s): Tohru Eguchi and Kazuhiro Hikami
  • Journal Title: Lett. Math. Phys. Volume: 102, no. 2 Pages: 203-222
  • Peer Reviewed
• [Journal Article] N=2 moonshine. (2012)
  • Author(s): Tohru Eguchi and Kazuhiro Hikami
  • Journal Title: Phys. Lett. B 717 Volume: no.1-3 Pages: 266-273
  • Peer Reviewed
• [Journal Article] Twisted Elliptic Genus and Borcherds Product (2012)
  • Author(s): T. Eguchi, K. Hikami
  • Journal Title: Lett. Math. Phys. Volume: 102 Issue: 2 Pages: 203-222
  • DOI: 10.1007/s11005-012-0569-2
  • Peer Reviewed
• [Journal Article] N=2 moonshine (2012)
  • Author(s): T. Eguchi, K. Hikami
  • Journal Title: Phys.Lett. B Volume: 717 Issue: 1-3 Pages: 266-273
  • DOI: 10.1016/j.physletb.2012.09.037
  • Peer Reviewed
• [Journal Article] Representations and the colored Jones polynomial of a torus knot. Chern-Simons gauge theory: 20 years after (2011)
  • Author(s): Kazuhiro Hikami and Hitoshi Murakami
  • Journal Title: AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI Volume: 50 Pages: 153-171
  • Peer Reviewed
• [Journal Article] Decomposition of Witten-Reshetikhin-Turaev invariant: linking pairing and modular forms. Chern-Simons gauge theory: 20 years after (2011)
  • Author(s): Kazuhiro Hikami
  • Journal Title: AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI Volume: 50, Amer. Math. Soc., Providence, RI Pages: 131-151
  • Peer Reviewed
• [Journal Article] On the modularity of the unified WRT invariants of certain Seifert manifolds. (2011)
  • Author(s): Kathrin Bringmann, Kazuhiro Hikami, and Jeremy Lovejoy
  • Journal Title: Adv. in Appl. Math. 46 Volume: no.1-4
  • Peer Reviewed
• [Journal Article] An introduction to the volume conjecture. Interactions between hyperbolic geometry, quantum topology and number theory (2011)
  • Author(s): Hitoshi Murakami
  • Journal Title: Contemp. Math., Amer. Math. Soc., Providence, RI Volume: 541 Pages: 1-40
  • Peer Reviewed
• [Journal Article] Representations and the colored Jones polynomial of a torus knot (2011)
  • Author(s): Hikami, Kazuhiro and Murakami, Hitoshi.
  • Journal Title: Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math. Volume: 50 Pages: 153-171
  • Peer Reviewed
• [Journal Article] An introduction to the volume conjecture (2011)
  • Author(s): Murakami, Hitoshi
  • Journal Title: Contemp. Math. Volume: 541 Pages: 1-40
  • Peer Reviewed
• [Journal Article] Decomposition of Witten-Reshetikhin-Turaev invariant: linking pairing and modular forms (2011)
  • Author(s): Hikami, Kazuhiro
  • Journal Title: Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math. Volume: 50 Pages: 131-151
  • Peer Reviewed
• [Journal Article] On the modularity of the unified WRT invariants of certain Seifert manifolds (2011)
  • Author(s): Bringmann, Kathrin, Hikami, Kazuhiro and Lovejoy, Jeremy
  • Journal Title: Adv. in Appl. Math. Volume: 46 Pages: 86-93
  • Peer Reviewed
• [Journal Article] Note on twisted elliptic genus of K3 surface (2011)
  • Author(s): Eguchi, Tohru and Hikami, Kazuhiro
  • Journal Title: Phys. Lett. B Volume: 694 Pages: 446-455
  • Peer Reviewed
• [Journal Article] An Introduction to the Volume Conjecture (2011)
  • Author(s): H.Murakami
  • Journal Title: Contemporary Mathematics "Interactions Between Hyperbolic Geometry Quantum Topology and Number Theory" Volume: 541(In Press)
  • Peer Reviewed
• [Journal Article] N=2Superconformal algebras and the entropy of Calabi-Yau manifolds (2010)
  • Author(s): Tohru Eguchi and Kazuhiro Hikami
  • Journal Title: Lett. Math. Phys.92 Volume: no. 3 Pages: 269-297
  • Peer Reviewed
• [Journal Article] N=4 superconfor mal algebra and the entropy of hyperKahler manifolds. (2010)
  • Author(s): Tohru Eguchi and Kazuhiro Hikami
  • Journal Title: J. High Energy Phys. Volume: no. 2 Pages: 19-46
  • Peer Reviewed
• [Journal Article] 体積予想の現状 (2010)
  • Author(s): 村上斉
  • Journal Title: 数学 Volume: 62巻 Pages: 502-523
  • NAID: 130004558929
  • Peer Reviewed
• [Journal Article] N=4 Superconformal algebras and the entropy of hyperKahler manifolds (2010)
  • Author(s): T.Eguchi, K.Hikami
  • Journal Title: Journal of High Energy Physics Volume: 2010 Pages: 19-46
  • Peer Reviewed
• [Journal Article] N=2 Superconformal algebras and the entropy of Calabi-Yau manifolds (2010)
  • Author(s): T.Eguchi, K.Hikami
  • Journal Title: N=2 Superconformal algebras and the entropy of Calabi-Yau manifolds Volume: 92 Pages: 269-297
  • Peer Reviewed
• [Journal Article] Representations and the colored Jones polynomial of a torus knot
  • Author(s): K.Hikami, H.Murakami
  • Journal Title: AMS/IP Studies in Advanced Mathematics publication- "Chern-Simons Theory : 20 years after" Volume: (In Press)
  • Peer Reviewed
• [Presentation] On the Complex Volume of Knots (2013)
  • Author(s): Kazuhiro Hikami
  • Organizer: quantum groups and beyond
  • Place of Presentation: Bethe Ansatz
  • Year and Date: 2013-03-09
• [Presentation] The volume conjecture and its generalizations (2013)
  • Author(s): Hitoshi Murakami
  • Organizer: Colloquium
  • Place of Presentation: University of Paris 7, Paris, France.
  • Year and Date: 2013-02-27
• [Presentation] Cluster algebra and complex volume of 2-bridge knots (2013)
  • Author(s): Kazuhiro Hikami
  • Organizer: Facets of integrability: random patterns, stochastic processes, hydrodynamics, gauge theories and condensed matter systems
  • Place of Presentation: Simons Center forGeometry and Physics, USA.
  • Year and Date: 2013-01-25
• [Presentation] The volume conjecture and its generalizations (2013)
  • Author(s): Hitoshi Murakami
  • Organizer: Colloquium
  • Place of Presentation: University of Paris 7, Paris, France
  • Invited
• [Presentation] Cluster algebra and complex volume of 2-bridge knots (2013)
  • Author(s): Kazuhiro Hikami
  • Organizer: Facets of integrability: random patterns, stochastic processes, hydrodynamics, gauge theories and condensed matter systems
  • Place of Presentation: Simons Center for Geometry and Physics, USA
  • Invited
• [Presentation] On the Complex Volume of Knots (2013)
  • Author(s): Kazuhiro Hikami
  • Organizer: Bethe Ansatz, quantum groups and beyond
  • Place of Presentation: RIMS, Kyoto
  • Invited
• [Presentation] Volume conjecture (2012)
  • Author(s): Hitoshi Murakami
  • Organizer: MPI-Oberseminar
  • Place of Presentation: Max Planck Institute forMathematics, Bonn, Germany.
  • Year and Date: 2012-09-06
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot (2012)
  • Author(s): Hitoshi Murakami
  • Organizer: Conference `New Perspectives in Topological Field Theories'
  • Place of Presentation: Center for Mathematical Physics, Hamburg, Germany.
  • Year and Date: 2012-08-31
• [Presentation] Decompositionof elliptic genera in terms of superconformal characters (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Low-Dimensional Topology and NumberTheory
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany.
  • Year and Date: 2012-08-31
• [Presentation] Superconformal algebra, mock theta functions, and moonshine, Hypergeometric Series and Their Generalizations in Algebra (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Geometry, Number Theory and Physics
  • Place of Presentation: Institut Henri Poincare, France.
  • Year and Date: 2012-05-29
• [Presentation] Moonshine from Character Decompositions of Elliptic Genera (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Workshop on Modular Forms
  • Place of Presentation: Kyushu University.
  • Year and Date: 2012-05-07
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot (2012)
  • Author(s): Hitoshi Murakami
  • Organizer: Conference `New Perspectives in Topological Field Theories'
  • Place of Presentation: Center for Mathematical Physics, Hamburg, Germany
  • Invited
• [Presentation] Volume conjecture (2012)
  • Author(s): Hitoshi Murakami
  • Organizer: MPI-Oberseminar
  • Place of Presentation: Max Planck Institute for Mathematics, Bonn, Germany
  • Invited
• [Presentation] Moonshine from Character Decompositions of Elliptic Genera (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Workshop on Modular Forms
  • Place of Presentation: Kyushu University
  • Invited
• [Presentation] Superconformal algebra, mock theta functions, and moonshine (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Hypergeometric Series and Their Generalizations in Algebra, Geometry, Number Theory and Physics
  • Place of Presentation: Institut Henri Poincare, France
  • Invited
• [Presentation] Decomposition of elliptic genera in terms of superconformal characters (2012)
  • Author(s): Kazuhiro Hikami
  • Organizer: Low-Dimensional Topology and Number Theory
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany
  • Invited
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot (2012)
  • Author(s): 村上 斉
  • Organizer: トポロジーセミナー
  • Place of Presentation: 東京工業大学
  • Invited
• [Presentation] The coloredJones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot (2011)
  • Author(s): Hitoshi Murakami
  • Organizer: Spring School in Geometry and Quantum Topology
  • Place of Presentation: LesDiablerets, Switzerland
  • Year and Date: 2011-03-25
• [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot (2011)
  • Author(s): H.Murakami
  • Organizer: Spring School in Geometry and Quantum Topology
  • Place of Presentation: Les Diablerets, Switzerland
  • Year and Date: 2011-03-25
• [Presentation] Superconformal algebra, mock theta function, and Mathieu moonshine (2011)
  • Author(s): Kazuhiro Hikami
  • Organizer: Workshop "Low dimensional topology and number theoryIII
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2011-03-15
• [Presentation] Superconformal algebra, mock theta function, and Mathieu moonshine (2011)
  • Author(s): K.Hikami
  • Organizer: Workshop "Low dimensional topology and number theory III"
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2011-03-15
• [Presentation] The colored Jones polynomial of a knot and its relation to the Chern-Simons invariant, and the Reidemeister torsion (2011)
  • Author(s): 村上 斉
  • Organizer: トポロジー金曜セミナー
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] Character decomposition of elliptic genus (2011)
  • Author(s): 樋上 和弘
  • Organizer: Workshop on Mathieu Moonshine
  • Place of Presentation: ETH Zurich （スイス）
  • Invited
• [Presentation] Volume conjecture and quantum dilogarithm function (2011)
  • Author(s): 樋上 和弘
  • Organizer: Simons Summer Workshop in Mathematics and Physics
  • Place of Presentation: Simons Center for Geometry and Physics （アメリカ）
  • Invited
• [Presentation] Ramanujan 擬テータ函数と Mathieu Moonshine (2011)
  • Author(s): 樋上 和弘
  • Organizer: 福岡数論
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] The colored Jones polynomial of a knot and representations of its undamental group (2010)
  • Author(s): Hitoshi Murakami
  • Organizer: Workshop `Low-Dimensional Topology and Number Theory'
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany.
  • Year and Date: 2010-08-20
• [Presentation] The colored Jones polynomial of a knot and representations of its fundamental group (2010)
  • Author(s): H.Murakami
  • Organizer: Workshop 'Low-Dimensional Topology and Number Theory'
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany
  • Year and Date: 2010-08-20
• [Presentation] Asymptotics ofQuantum Invariants (2010)
  • Author(s): Kazuhiro Hikami
  • Organizer: Workshop`Low-Dimensional Topology and NumberTheory'
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany.
  • Year and Date: 2010-08-19
• [Presentation] Asymptotics of Quantum Invariants (2010)
  • Author(s): K.Hikami
  • Organizer: Workshop 'Low-Dimensional Topology and Number Theory'
  • Place of Presentation: Mathematical Research Institute of Oberwolfach, Germany
  • Year and Date: 2010-08-19
• [Presentation] Superconformal Algebras and the Entropy of HyperKahler Manifolds (2010)
  • Author(s): Kazuhiro Hikami
  • Organizer: Workshop "Prospects in q-series and modular forms
  • Place of Presentation: University College Dublin, Ireland.
  • Year and Date: 2010-07-14
• [Presentation] Superconformal Algebras and the Entropy of HyperKahler Manifolds (2010)
  • Author(s): K.Hikami
  • Organizer: Workshop "Prospects in q-series and modular forms"
  • Place of Presentation: University College Dublin, Ireland
  • Year and Date: 2010-07-14