Study on geometry of knots and three manifolds by using representation theory of quantum groups

• Project/Area Number: 19540230
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Waseda University
• Principal Investigator: MURAKAMI Jun Waseda University, 理工学術院, 教授 (90157751)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2007: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 量子群 / 結び目 / 3次元多様体 / 表現論 / TQFT / 結び目理論 / 組紐群

Research Abstract

The volume conjecture for the quantum invariants of knots is studied from the view point of the colored Alexander invariant, and the logarithmic invariants and SL (2, C) quantum 6j symbols are constructed. The relation between these and the hyperbolic volume is given.

Research Products

• [Journal Article] Some limits of the colored Alexander invariant of the figure-eight knot and the volume of hyperbolic orbifolds. (2009)
  • Author(s): Jinseok Cho, Jun Murakami
  • Journal Title: Journal of Knot Theory and its Ramifications Vol.18,no.9 Pages: 1271-1286
  • Peer Reviewed
• [Journal Article] The complex volumes of twist knots. (2009)
  • Author(s): Jinseok Cho, Jun Murakami, Yoshiyuki Yokota
  • Journal Title: Proceedings of the American Mathematical Society Vol.137,no.10 Pages: 3533-3541
  • Peer Reviewed
• [Journal Article] Some limits of the colored Alexander invariant of the figure-eight knot and the volume of hyperbolic orbifolds (2009)
  • Author(s): Cho, Jinseok, Murakami, Jun
  • Journal Title: J.Knot Theory Ramifications 18 Pages: 1271-1286
  • Peer Reviewed
• [Journal Article] The complex volumes of twist knots (2009)
  • Author(s): Cho, Jinseok, Murakami, Jun, Yokota, Yoshiyuki
  • Journal Title: Proceedings of American Mathematical Society 137 Pages: 3533-3541
  • Peer Reviewed
• [Journal Article] 量子不変量と双曲体積の思いがけない対応 (2009)
  • Author(s): 村上順
  • Journal Title: 数理科学 47 Pages: 13-18
• [Journal Article] Logarithmic knot invariants arising from restricted quantum groups. (2008)
  • Author(s): Jun Murakami, Kiyokazu Nagatomo
  • Journal Title: International Journal of Mathematics Vol.19,no.10 Pages: 1203-1213
  • Peer Reviewed
• [Journal Article] Colored Alexander invariants and cone-manifolds. (2008)
  • Author(s): Jun Murakami
  • Journal Title: Osaka Journal of Mathematics Vol.45,no.2 Pages: 541-564
  • NAID: 120004843322
  • Peer Reviewed
• [Journal Article] Logarithmic knot invariants arising from restricted quantum groups (2008)
  • Author(s): Murakami, Jun, Nagatomo, Kiyokazu
  • Journal Title: International Journal of Mathematics 19 Pages: 1203-1213
  • Peer Reviewed
• [Journal Article] Colored Alexander invariants and oone-manifolds (2008)
  • Author(s): Murakami, Jun
  • Journal Title: Osaka Journal of Mathematics 45 Pages: 541-564
  • Peer Reviewed
• [Journal Article] Colored Alexander invariants and cone-manifolds
  • Author(s): Jun Murakami
  • Journal Title: Osaka Journal of Mathematics accepted
  • NAID: 120004843322
  • Peer Reviewed
• [Journal Article] Logarithicmic knot invariants arising from resricted quantum groups
  • Author(s): Jun Murakami and Kiyokazu Nagatomo
  • Journal Title: International Journal of Mathematics accepted
  • Peer Reviewed
• [Presentation] On the tensor category of projective modules of the small quantum groups (2009)
  • Author(s): 村上順
  • Organizer: Category Theory, Computer Science, and Topology
  • Place of Presentation: 信州大学
  • Year and Date: 2009-10-18
• [Presentation] On the quautum 6j symbols for the non-integral highest weight representations of Uq(sl2) at root of 1 (2009)
  • Author(s): 村上順
  • Organizer: 2009年度日本数学会秋期総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] On the quantum 6j symbols for the non-integral highest weight representations of U_q(sl_2)at root of 1 (2009)
  • Author(s): 村上順
  • Organizer: 日本数学会秋期総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] On the extra quantum group U_\xi^(r)(sl_2) at root of 1 (2009)
  • Author(s): 村上順
  • Organizer: 2009年度日本数学会総会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-28
• [Presentation] On the extra quantum group U_\xi^(r)(sl_2) at root of 1 (2009)
  • Author(s): 村上 順
  • Organizer: 2009年度日本数学会年会
  • Place of Presentation: 東京大学駒場キャンパス
  • Year and Date: 2009-03-28
• [Presentation] On the variations of quantum sl2 invariants of knots and links (2009)
  • Author(s): 村上順
  • Organizer: Workshop on Turaev-Viro invariant and related topics
  • Place of Presentation: 東京工業大学
  • Year and Date: 2009-02-10
• [Presentation] On the variations of quan-tum sl2 invariants of knots and links (2009)
  • Author(s): Murakami, Jun
  • Organizer: Workshop on Turaev-Viro invariant and related topics
  • Place of Presentation: 東京工業大学大岡山キャンパス
  • Year and Date: 2009-02-10
• [Presentation] On logarithmic knot invariant (2008)
  • Author(s): Jun Murakami
  • Organizer: Braids in Paris
  • Place of Presentation: Univ. Paris Diderot (Paris 7), FRANCE
  • Year and Date: 2008-09-18
• [Presentation] On logarithmic knot invariant (2008)
  • Author(s): Murakami, Jun
  • Organizer: Braids in Paris
  • Place of Presentation: Univ. Paris Diderot (Paris 7)
  • Year and Date: 2008-09-18
• [Presentation] On branch problem of complex volume (2008)
  • Author(s): 村上順
  • Organizer: Topology and Computer 2008
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-08-23
• [Presentation] On branch problem of complex volume (2008)
  • Author(s): Murakami, Jun
  • Organizer: トポロジーとコンピュータ2008
  • Place of Presentation: 東京工業大学大岡山キャンパス
  • Year and Date: 2008-08-23
• [Presentation] On the TQFT Coming from the Restricted Quantum Group Finite Type Invariants (2008)
  • Author(s): Jun Murakami
  • Organizer: Fat Graphs and Torelli-Johnson-Morita Theory
  • Place of Presentation: CTQM, University of Aarhus, DENMARK
  • Year and Date: 2008-04-03
• [Presentation] On the TQFT Coining from the Restricted Quantum Group (2008)
  • Author(s): Murakami, Jun
  • Organizer: Finite Type Invariants, Fat Graphs and Torelli-Johnson-Morita Theory
  • Place of Presentation: CTQM, University of Aarhus
  • Year and Date: 2008-04-03
• [Presentation] On logarithmic knot invariant (2008)
  • Author(s): Jun Murakami
  • Organizer: 4th East Asia School of Knots, Links and Related Topics
  • Place of Presentation: 東京大学
  • Year and Date: 2008-01-23
• [Presentation] 量子不変量の様々なstate model, 量子6j-symbolと双曲四面体の体積, Colored Alexander不変量と体積予想 (2008)
  • Author(s): 村上順
  • Organizer: 夏の学校「低次元トポロジーにおける未解決問題」
  • Place of Presentation: 名古屋大学
• [Presentation] Uq(sl2)と関連する写像類群の表現について (2007)
  • Author(s): 村上順
  • Organizer: Workshop on linear representations of the mapping class group and related topics
  • Place of Presentation: 高知県, 国民宿舎桂浜荘
  • Year and Date: 2007-12-22
• [Presentation] On representations of mapping class groups related to Uq(s12) (2007)
  • Author(s): Jun Murakami
  • Organizer: Workshop on linear representations of the mapping class group and related topics
  • Place of Presentation: 高知桂浜荘
  • Year and Date: 2007-12-22
• [Presentation] Uq(sl2) at qN=1と2+1次元トポロジー (2007)
  • Author(s): 村上順
  • Organizer: 2+1次元トポロジーの新しい流れ
  • Place of Presentation: 早稲田大学
  • Year and Date: 2007-11-14
• [Presentation] Uq(s12)at qn=1と2+1次元トポロジー (2007)
  • Author(s): Jun Murakami
  • Organizer: 2+1次元トポロジーの新しい流れ
  • Place of Presentation: 早稲田大学
  • Year and Date: 2007-11-14
• [Presentation] Colored Alexander invariant and cone manifolds (2007)
  • Author(s): Jun Murakami
  • Organizer: Workshop Hyperbolic Volume
  • Place of Presentation: Universite Fribourg, SWISS
  • Year and Date: 2007-07-03
• [Presentation] Colored Alexander invariant and cone manifold (2007)
  • Author(s): Jun Murakami
  • Organizer: workshop Hyperbolic Volumes 2007
  • Place of Presentation: Fribourg,スイス
  • Year and Date: 2007-07-03
• [Remarks]
  • URL: http://www.f.waseda.jp/murakami/
• [Remarks]
  • URL: http://www.f.waseda.jp/murakami
• [Remarks]
  • URL: http://www.f.waseda.jp/murakami/researoh-jp.html