Solving the unknotting conjecture in dimension four and its development

• Project/Area Number: 21540084
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Hiroshima University
• Principal Investigator: MATUMOTO Takao 広島大学, 大学院理学研究科, 名誉教授 (50025467)
• Co-Investigator(Kenkyū-buntansha): KAMADA Seiichi 広島大学, 大学院理学研究科, 教授 (60254380)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ２次元結び目 / ２次元ブレイド / ４次元トポロジー / 数学史 / 2次元結び目 / 2次元ブレイド / チャート図の変形 / マルコフ型定理 / 4次元トポロジー / ケーリー図 / 和算と行列式 / 不変量 / 和算と行列式・ベルヌーイ数 / カスプ / 4次トポロジー

Research Abstract

The smooth unknotting conjecture in dimension four is reduced to the case which is connected by a one-parameter family with one intersection point to the unknot, by assuming our Markov type theorem. Moreover, since the reduced case can be studied by a kind of tree type Cayley diagram for the corresponding word problem, the conjecture is expected to be proved in a near future.

Research Products

• [Journal Article] On the smooth unknotting conjecture in dimension four, Intelligence of Low-dimensional Topology (2011)
  • Author(s): Takao Matumoto
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1766 Pages: 1-14
  • NAID: 110008677032
• [Journal Article] Complement to explicit description of Hopf surfaces and their automorphism groups (2011)
  • Author(s): Takao Matumoto and Noriaki Nakagawa
  • Journal Title: Osaka J. Math Volume: 48 Pages: 581-588
  • NAID: 120004842860
  • Peer Reviewed
• [Journal Article] Bridge presentations of virtual knots (2011)
  • Author(s): Mikami Hirasawa, Naoko Kamada and Seiichi Kamada
  • Journal Title: J. Knot Theory and its Ramifications Volume: 20 Pages: 881-893
  • Peer Reviewed
• [Journal Article] 関の逐式交乗表の解釈とその先 (2011)
  • Author(s): 松本堯生
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1739 Pages: 102-113
• [Journal Article] On the smooth unknotting conjecture in dimension four (2011)
  • Author(s): Takao Matumoto
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1766 Pages: 1-14
• [Journal Article] Complement to explicit description of Hopf surfaces and their automorphism groups (2011)
  • Author(s): Takao Matumoto, N.Nakagawa
  • Journal Title: Osaka Journal of Mathematics Volume: 48 Pages: 583-588
  • NAID: 120004842860
  • Peer Reviewed
• [Journal Article] Bridge presentations of virtual knots (2011)
  • Author(s): M. Hirasawa, N. Kamada and S. Kamada
  • Journal Title: J. Knot Theory Ramifications Volume: 20 Issue: 06 Pages: 881-893
  • DOI: 10.1142/s0218216511009017
  • Peer Reviewed
• [Journal Article] Complement to explicit description of Hopf surfces and their automorphism groups (2011)
  • Author(s): 松本堯生・中川規明
  • Journal Title: Osaka Journal of Mathematics Volume: 48(To appear)
  • Peer Reviewed
• [Journal Article] The Miyazawa polynomial for long virtual knots (2010)
  • Author(s): Atsushi Ishii, Naoko Kamada and Seiichi Kamada
  • Journal Title: Topology Appl Volume: 157 Pages: 290-297
  • Peer Reviewed
• [Journal Article] Quandles derived from dynamical systems and subsets which are closed under quandle operations (2010)
  • Author(s): Seiichi Kamada
  • Journal Title: Topology Appl Volume: 157 Pages: 298-301
  • Peer Reviewed
• [Journal Article] Homology groups of symmetric quandles and cocycle invariants of links and surface-links (2010)
  • Author(s): Seiichi Kamada and Kanako Oshiro
  • Journal Title: Trans. Amer. Math. Soc Volume: 362 Pages: 5501-5527
  • Peer Reviewed
• [Journal Article] New invariants of long virtual knots (2010)
  • Author(s): Andrew Bartholomew, Roger Fenn, Naoko Kamada and Seiichi Kamada
  • Journal Title: Kobe J. Math Volume: 27 Pages: 21-33
  • Peer Reviewed
• [Journal Article] Homology groups of symmetric quandles and cocycle invariants of links and surface-linkas (2010)
  • Author(s): 鎌田聖一・大城佳奈子
  • Journal Title: Transactions of American Mathematical Society Volume: 362 Pages: 5501-5527
  • Peer Reviewed
• [Journal Article] Quandles derived from dynamical systems and subsets which are closed under quandle operations (2010)
  • Author(s): 鎌田聖一
  • Journal Title: Topology and its Applications Volume: 157 Pages: 298-331
  • Peer Reviewed
• [Journal Article] Quandles derived from dynamical systems and subsets which are closed under quandle operations (2010)
  • Author(s): 鎌田聖一
  • Journal Title: Topology and its Applications 157 Pages: 298-301
  • Peer Reviewed
• [Journal Article] The Miyazawa polynomial for long virtual knots (2010)
  • Author(s): 鎌田聖一, 他
  • Journal Title: Topology and its Applications 157 Pages: 290-297
  • Peer Reviewed
• [Journal Article] Sanpo-hakki 算法発揮 the first published book on determinants and related works by Seki and others (2010)
  • Author(s): 松本堯生
  • Journal Title: Proc.History of Mathematics of Seki Takakazu (1642?-1708) Tokyo 2008 (To appear)
  • Peer Reviewed
• [Presentation] On the smooth unknotting conjecture in dimension four VII (2011)
  • Author(s): Takao Matumoto
  • Organizer: 4次元トポロジー
  • Place of Presentation: 広島大学
  • Year and Date: 2011-11-09
• [Presentation] Biquandles with structures related to virtual links and twisted links (2011)
  • Author(s): Seiichi Kamada
  • Organizer: The 19th TAPU Seminar on Knots and Related Topics
  • Place of Presentation: Seacloud Hotel, 釜山, 韓国
  • Year and Date: 2011-09-15
• [Presentation] On the smooth unknotting conjecture in dimension four (2011)
  • Author(s): Takao Matumoto
  • Organizer: Intelligence of Low-dimensional Topology
  • Place of Presentation: RIMS, Kyoto Univ
  • Year and Date: 2011-05-25
• [Presentation] Monodromy representations and their graphical descriptions (2011)
  • Author(s): Seiichi Kamada
  • Organizer: Workshop on Low Dimensional Topology in Shanghai and Suzhou
  • Place of Presentation: 蘇州大学, 上海, 中国
  • Year and Date: 2011-05-17
• [Presentation] On the smooth unknotting conjecture in dimension four (2011)
  • Author(s): 松本堯生
  • Organizer: Intelligence of Low-Dimensional Topology
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2011-05-16
• [Presentation] ガリレオとガロア(旧世界からの脱皮) (2011)
  • Author(s): 松本堯生
  • Organizer: 関孝和数学研究所3.14…数学文化シンポジウム
  • Place of Presentation: 名城大学
  • Year and Date: 2011-03-13
• [Presentation] Genus one Lefschetz fibrations with a certain non-trivial global monodromy (2011)
  • Author(s): Seiichi Kamada
  • Organizer: Seventh East Asian School of Knots and Related Topics
  • Place of Presentation: 広島大学
  • Year and Date: 2011-01-12
• [Presentation] On the smooth unknotting conjecture in dimension four VI (2010)
  • Author(s): Takao Matumoto
  • Organizer: 4次元トポロジー
  • Place of Presentation: 広島大学
  • Year and Date: 2010-11-15
• [Presentation] On the unknotting conjecture in dimension four VI (2010)
  • Author(s): 松本堯生
  • Organizer: 4次元トポロジー
  • Place of Presentation: 広島大学(招待講演)
  • Year and Date: 2010-11-15
• [Presentation] 関の逐式交乗表の解釈とその先 (2010)
  • Author(s): 松本堯生
  • Organizer: 数学史の研究
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-08-23
• [Presentation] 和算における行列式五種 (2010)
  • Author(s): 松本堯生
  • Organizer: 日本数学史学会
  • Place of Presentation: 同志社大学
  • Year and Date: 2010-05-23
• [Presentation] On quandles and biquandles related to twisted virtual links (2010)
  • Author(s): Seiichi Kamada
  • Organizer: Knots in Washington XXX
  • Place of Presentation: George Washington University, 米国
  • Year and Date: 2010-05-20
• [Presentation] On symmetric quandle presentations and surface-knots (2010)
  • Author(s): Seiichi Kamada
  • Organizer: The Sixth East Asian School of Knots and Related Topics
  • Place of Presentation: 南開大学, 天津, 中国
  • Year and Date: 2010-01-26
• [Presentation] On the smooth unknotting conjecture in dimension four V (2010)
  • Author(s): Takao Matumoto
  • Organizer: 4次元トポロジー
  • Place of Presentation: 広島大学
  • Year and Date: 2010-01-20
• [Presentation] On the unknotting conjecture in dimension four V (2010)
  • Author(s): 松本堯生
  • Organizer: 4次元トポロジー研究集会
  • Place of Presentation: 広島大学(東広島)
  • Year and Date: 2010-01-20
• [Presentation] Knot symmetric quandles and their presentations (2009)
  • Author(s): Seiichi Kamada
  • Organizer: アメリカ数学会 Fall Western Sectional Meeting
  • Place of Presentation: Algebraic Structure in Knot Theory, Univ. California at Riverside, 米国
  • Year and Date: 2009-11-08
• [Presentation] 結び目のトポロジーとカンドル代数 (2009)
  • Author(s): 鎌田聖一
  • Organizer: 日本数学会総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-26
• [Book] 微分トポロジー(ハーシュ著Differential Topologyの訳書) (2012)
  • Author(s): 松本堯生
  • Publisher: 丸善 (2012出版確定)
• [Book] 曲面結び目理論 (2012)
  • Author(s): 鎌田聖一
  • Publisher: 丸善
• [Book] 微分トポロジー(ハーシュ著) (2012)
  • Author(s): 松本堯生(訳者)
  • Total Pages: 280
  • Publisher: 丸善株式会社(編集はシュプリンガージャパン)