Categorification of quantum invariants of 3-manifolds

• Project/Area Number: 23740062
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Waseda University
• Principal Investigator: ITO Noboru 早稲田大学, 理工学術院, 助教 (10580160)
• Project Period (FY): 2011 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2011: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
• Keywords: 低次元トポロジー / 量子トポロジー / 結び目 / ３次元多様体 / 量子不変量 / ２重複体 / スペクトル系列 / 量子化 / 量子群 / 国際研究者交流（デンマーク） / 国際研究者交流（アメリカ）

Research Abstract

This study gave a fundamental theory contributing to attack the famous open problem: "Categorificaiton of quantum invariants of 3-manifolds". The key point to solve the problem is that we consider the possibility to define invariants under two kinds of operations, denoted by K1 and K2. The main three results are as follows. (1) We reveal that K1 corresponds to sliding grades of a known categorification. (2) We obtain the objects that are invariant under K2 by redefining a generalization of known categorification. (3) We gave an explicit description of a geometric operation derived from (2).

Research Products

• [Journal Article] Survey and Remarks on Viro's definition of Khovanov homology (2013)
  • Author(s): Noboru Ito
  • Journal Title: RIMS Kokyuroku, Bessatsu Volume: B39 Pages: 17-17
  • Peer Reviewed
• [Journal Article] Jones polynomials of long virtual knots (2013)
  • Author(s): Noboru Ito
  • Journal Title: J. Knot Theory Ramifications Volume: 22 Pages: 17-17
  • Peer Reviewed
• [Journal Article] Survey and Remarks on Viro's definition of Khovanov homology (2013)
  • Author(s): Noboru Ito
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B39
  • Peer Reviewed
• [Journal Article] The half-twisted splice operation on reduced knot projections (2012)
  • Author(s): Ayaka Shimizu, Noboru Ito
  • Journal Title: J. Knot Theory Ramifications Volume: 21 Pages: 10-10
  • Peer Reviewed
• [Journal Article] The half-twisted splice operation on reduced knot projections (2012)
  • Author(s): Aayaka Shimizu
  • Journal Title: J. Knot Theory Ramifications Volume: 21 Issue: 11 Pages: 1250112-1250112
  • DOI: 10.1142/s021821651250112x
  • Peer Reviewed
• [Presentation] A case study on WRT invariant and Khovanov homology (2013)
  • Organizer: Special Topology Seminar, Department of Mathematics
  • Place of Presentation: The George Washington University
• [Presentation] A case of the WRT invariant and Khovanov homology (2013)
  • Organizer: Knots in Washington XXXVI
  • Place of Presentation: The George Washington University
• [Presentation] Khovanovホモロジー入門1 (2013)
  • Author(s): 伊藤昇
  • Organizer: Categorification勉強会2013
  • Place of Presentation: 東京大学
• [Presentation] Khovanovホモロジー入門2 (2013)
  • Author(s): 伊藤昇
  • Organizer: Categorification勉強会2013
  • Place of Presentation: 東京大学
• [Presentation] Khovanovホモロジー入門3 (2013)
  • Author(s): 伊藤昇
  • Organizer: Categorification勉強会2013
  • Place of Presentation: 東京大学
• [Presentation] Introduction to non-triviality of Gauss words (2012)
  • Author(s): Noboru Ito
  • Organizer: The Discrete Mathematics Seminar
  • Place of Presentation: The George Washington University
• [Presentation] Khovanov homology and Kirby moves (2012)
  • Author(s): Noboru Ito
  • Organizer: Knots in Washington XXXV
  • Place of Presentation: The George Washington University
• [Presentation] Invariants of curves on surfaces and knot invariants (2012)
  • Author(s): 伊藤昇
  • Organizer: リーマン面に関連する位相幾何学
  • Place of Presentation: 東京大学
• [Presentation] On Khovanov bicomplex and related problems (2012)
  • Author(s): Noboru Ito
  • Organizer: Topology Seminar, Department of Mathematics
  • Place of Presentation: The George Washington University
• [Presentation] Half-twisted splice on reduced knot projections (2012)
  • Author(s): Noboru Ito
  • Organizer: Topology Seminar, Department of Mathematics
  • Place of Presentation: University of California, Santa Barbara
• [Presentation] Half-twisted splice on reduced knot projections (2012)
  • Author(s): Noboru Ito
  • Organizer: Topology Seminar(招待講演)
  • Place of Presentation: Department of Mathematics, University of California, Santa Barbara
• [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases (2011)
  • Author(s): 伊藤昇
  • Organizer: コロキウム
  • Place of Presentation: 立教大学
• [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases (2011)
  • Author(s): 伊藤昇
  • Organizer: 信州トポロジーセミナー
  • Place of Presentation: 信州大学
• [Presentation] Remarks on nanophrases and Jones polynomials (2011)
  • Author(s): 伊藤昇
  • Organizer: 東北結び目セミナー
  • Place of Presentation: 東北大学
• [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant (2011)
  • Author(s): 伊藤昇
  • Organizer: 第58回トポロジーシンポジウム
  • Place of Presentation: 筑波大学
• [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant and 3-manifolds (2011)
  • Author(s): Noboru Ito
  • Organizer: Geometry of Transformation Groups and Combinatorics, RIMS
  • Place of Presentation: Kyoto University
• [Presentation] A relation between Khovanov homology and Kirby moves (2011)
  • Author(s): Noboru Ito
  • Organizer: Knots in Washington XXXII
  • Place of Presentation: The George Washington University
• [Presentation] A relation between Khovanov homology and Kirby moves (2011)
  • Author(s): Noboru Ito
  • Organizer: Knots in Washington XXXII(招待講演)
  • Place of Presentation: The George Wshington University
• [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant and 3-manifolds (2011)
  • Author(s): Noboru Ito
  • Organizer: Geometry of Transformation Groups and Combinatrics
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant (2011)
  • Author(s): Noboru Ito
  • Organizer: The 58th Topology Symposium(招待講演)
  • Place of Presentation: 筑波大学
• [Presentation] Remarks on nanophrases and Jones polynomials (2011)
  • Author(s): Noboru Ito
  • Organizer: Tohoku Knot Seminar
  • Place of Presentation: 東北大学
• [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases (2011)
  • Author(s): Noboru Ito
  • Organizer: Shinshu Topology Seminar(招待講演)
  • Place of Presentation: 信州大学
• [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases (2011)
  • Author(s): Noboru Ito
  • Organizer: Rikkyo Univeristy(招待講演)
  • Place of Presentation: 立教大学
• [Presentation] Khovanov homology入門１
  • Author(s): 伊藤昇
  • Organizer: Categorificaiton勉強会2013
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Khovanov homology入門２
  • Author(s): 伊藤昇
  • Organizer: Categorification勉強会2013
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Khovanov homology入門３
  • Author(s): 伊藤昇
  • Organizer: Categorification勉強会2013
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Introduction to non-triviality of Gauss words
  • Author(s): Noboru Ito
  • Organizer: The Discrete Mathematics Seminar
  • Place of Presentation: University of South Florida
  • Invited
• [Presentation] Introduction to Khovanov homology and Kirby moves
  • Author(s): Noboru Ito
  • Organizer: Topology Seminar, Department of mathematics
  • Place of Presentation: The George Washington University
  • Invited
• [Presentation] On Khovanov bicomplex and related problems
  • Author(s): Noboru Ito
  • Organizer: Topology Seminar, Department of Mathematics
  • Place of Presentation: The George washington University
  • Invited
• [Remarks] 伊藤昇・谷山公規、研究集会 「結び目の量子不変量とそのカテゴリー化」、早稲田大学早稲田キャンパス、2011年8月24日-8月26日
  • URL: http://www.waseda.jp/prj-knot_quantum/index.html
• [Remarks]
  • URL: http://www.aoni.waseda.jp/noboru/
• [Remarks] Noboru Ito's page
  • URL: http://www.aoni.waseda.jp/noboru/
• [Remarks]
  • URL: http://www.aoni.waseda.jp/noboru/index.html