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2012 Fiscal Year Final Research Report

Categorification of quantum invariants of 3-manifolds

Research Project

  • PDF
Project/Area Number 23740062
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionWaseda University

Principal Investigator

ITO Noboru  早稲田大学, 理工学術院, 助教 (10580160)

Project Period (FY) 2011 – 2012
Keywords低次元トポロジー
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

    (21 results)

All 2013 2012 2011 Other

All Journal Article (3 results) (of which Peer Reviewed: 3 results) Presentation (16 results) Remarks (2 results)

  • [Journal Article] Survey and Remarks on Viro's definition of Khovanov homology2013

    • Author(s)
      Noboru Ito
    • Journal Title

      RIMS Kokyuroku, Bessatsu

      Volume: B39 Pages: 17

    • Peer Reviewed
  • [Journal Article] Jones polynomials of long virtual knots2013

    • Author(s)
      Noboru Ito
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 22 Pages: 17

    • Peer Reviewed
  • [Journal Article] The half-twisted splice operation on reduced knot projections2012

    • Author(s)
      Ayaka Shimizu, Noboru Ito
    • Journal Title

      J. Knot Theory Ramifications

      Volume: 21 Pages: 10

    • Peer Reviewed
  • [Presentation] A case study on WRT invariant and Khovanov homology2013

    • Organizer
      Special Topology Seminar, Department of Mathematics
    • Place of Presentation
      The George Washington University
    • Year and Date
      20130500
  • [Presentation] A case of the WRT invariant and Khovanov homology2013

    • Organizer
      Knots in Washington XXXVI
    • Place of Presentation
      The George Washington University
    • Year and Date
      20130500
  • [Presentation] Khovanovホモロジー入門12013

    • Author(s)
      伊藤昇
    • Organizer
      Categorification勉強会2013
    • Place of Presentation
      東京大学
    • Year and Date
      20130300
  • [Presentation] Khovanovホモロジー入門22013

    • Author(s)
      伊藤昇
    • Organizer
      Categorification勉強会2013
    • Place of Presentation
      東京大学
    • Year and Date
      20130300
  • [Presentation] Khovanovホモロジー入門32013

    • Author(s)
      伊藤昇
    • Organizer
      Categorification勉強会2013
    • Place of Presentation
      東京大学
    • Year and Date
      20130300
  • [Presentation] Introduction to non-triviality of Gauss words2012

    • Author(s)
      Noboru Ito
    • Organizer
      The Discrete Mathematics Seminar
    • Place of Presentation
      The George Washington University
    • Year and Date
      20121200
  • [Presentation] Khovanov homology and Kirby moves2012

    • Author(s)
      Noboru Ito
    • Organizer
      Knots in Washington XXXV
    • Place of Presentation
      The George Washington University
    • Year and Date
      20121200
  • [Presentation] Invariants of curves on surfaces and knot invariants2012

    • 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
    • Year and Date
      20120900
  • [Presentation] On Khovanov bicomplex and related problems2012

    • Author(s)
      Noboru Ito
    • Organizer
      Topology Seminar, Department of Mathematics
    • Place of Presentation
      The George Washington University
    • Year and Date
      20120500
  • [Presentation] Half-twisted splice on reduced knot projections2012

    • Author(s)
      Noboru Ito
    • Organizer
      Topology Seminar, Department of Mathematics
    • Place of Presentation
      University of California, Santa Barbara
    • Year and Date
      20120300
  • [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases2011

    • Author(s)
      伊藤昇
    • Organizer
      コロキウム
    • Place of Presentation
      立教大学
    • Year and Date
      20111100
  • [Presentation] Chain homotopy maps for Khovanov homology and finite type invariants of nanophrases2011

    • Author(s)
      伊藤昇
    • Organizer
      信州トポロジーセミナー
    • Place of Presentation
      信州大学
    • Year and Date
      20111100
  • [Presentation] Remarks on nanophrases and Jones polynomials2011

    • Author(s)
      伊藤昇
    • Organizer
      東北結び目セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      20111000
  • [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant2011

    • 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
    • Year and Date
      20110800
  • [Presentation] Khovanov homology for an unnormalized Witten-Reshetikhin-Turaev invariant and 3-manifolds2011

    • Author(s)
      Noboru Ito
    • Organizer
      Geometry of Transformation Groups and Combinatorics, RIMS
    • Place of Presentation
      Kyoto University
    • Year and Date
      20110600
  • [Presentation] A relation between Khovanov homology and Kirby moves2011

    • Author(s)
      Noboru Ito
    • Organizer
      Knots in Washington XXXII
    • Place of Presentation
      The George Washington University
    • Year and Date
      20110400
  • [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/

URL: 

Published: 2014-09-25  

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