Multi-valued Morse theory for knots and its applications

• Project/Area Number: 18540072
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Agriculture and Technology
• Principal Investigator: GODA Hirosh Tokyo University of Agriculture and Technology, 大学院・共生科学技術研究院, 教授 (60266913)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥2,460,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥360,000); Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 微分トポロジ- / 幾何学 / トポロジー / 結び目 / モース理論

Research Abstract

結び目補空間の多重値モース理論に付随するヘガード分解が誘導するチェイン複体と本来のNovikov 複体がチェインホモトピー同値であることを証明し, このヘガード分解を使ったトーションおよびゼータ関数の計算方法を確立した. Homology cylinderという概念の結び目理論における対応物がHomological fibered knotであるということを示した. Homology cylinderに対して定義されたトーションを利用して, 結び目がファイバー結び目になる為の障害定理を作った.

Research Products

• [Journal Article] Dynamics of gradient flows in the half-transversal Morse theory (2009)
  • Author(s): Hiroshi Goda, Andrei Pajitnov
  • Journal Title: Proceedings of the Japan Academy, Ser.A 85 Pages: 6-10
  • NAID: 40016423745
  • Peer Reviewed
• [Journal Article] Dynamics of gradient flows in the half-transversal Morse theory (2009)
  • Author(s): Hiroshi Goda, Andrei Pajitnov
  • Journal Title: Proceedings of the Japan Academy, Ser. A 85
  • NAID: 40016423745
  • Peer Reviewed
• [Journal Article] Some estimates of the Morse-Novikov numbers for knots and links (2007)
  • Author(s): Hiroshi Goda
  • Journal Title: Series on Knots and Everything 40 Pages: 35-42
  • Peer Reviewed
• [Journal Article] A classification of Seifert surfaces of some pretzel links (2006)
  • Author(s): Hiroshi Goda, Makiko Ishiwata
  • Journal Title: Kobe Journal of Mathematics 23 Pages: 11-28
  • Peer Reviewed
• [Presentation] Homological fibered knots and the Nakanishi index (2008)
  • Author(s): 合田洋
  • Organizer: 研究集会「Intelligence of Low dimensional Topology」兼拡大KOOKセミナー
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2008-10-07
• [Presentation] Homological fibered knots and the Nakanishi index (2008)
  • Author(s): 合田 洋
  • Organizer: 研究集会「Intelligence of Low dimensional. Topology」兼「拡大K00Kセミナー」
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2008-10-07
• [Presentation] Homology cylinders in knot theory (2008)
  • Author(s): Hiroshi Goda
  • Organizer: Seminaire de Topologie, Geometrie et.al.
  • Place of Presentation: Nantes, France
  • Year and Date: 2008-09-24
• [Presentation] Homology cylinders in knot theory (2008)
  • Author(s): Hiroshi Goda
  • Organizer: Seminaire de Topologie, Geometrie et Algebr
  • Place of Presentation: Universite de Nantes, France
  • Year and Date: 2008-09-24
• [Presentation] A fibering obstruction using the Magnus representation (2008)
  • Author(s): 合田洋
  • Organizer: 研究集会トポロジーとコンピュータ2008
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-08-28
• [Presentation] Morse-Novikov theory (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Heegaard splitting, and closed orbits of gradient flows, Seminaire d'Algebre et de Topologie
  • Place of Presentation: Nantes, France
  • Year and Date: 2007-09-13
• [Presentation] Morse-Novikov theory, Heegaard splittings, and closed orbits of gradient flows (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Seminaire d'Algebre et de Topologie
  • Place of Presentation: Nantes, France
  • Year and Date: 2007-09-13
• [Presentation] Morse-Novikov theory (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Heegaard splittinhs, and closed orbits of gradient flows, Braids, groups and manifolds in Toulouse
  • Place of Presentation: Toulouse, France
  • Year and Date: 2007-09-07
• [Presentation] Morse-Novikov theory, Heegaard splittings, and closed orbits of gradient flows (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Braids, groups and manifolds in Toulouse
  • Place of Presentation: Toulouse, France
  • Year and Date: 2007-09-07
• [Presentation] Circle-valued Morse maps, Reidemeister torsions, and sutured manifold theory (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Algebraic Topology : Old and New
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2007-06-18
• [Presentation] Circle valued Morse maps, Reidemeister torsions, and sutured manifold theory (2007)
  • Author(s): Hiroshi Goda
  • Organizer: Algebraic Tbpology: Old and New
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2007-06-18
• [Presentation] ねじれAlexander不変量とその周辺 (2007)
  • Author(s): 合田洋
  • Organizer: 金沢大学談話会
  • Place of Presentation: 金沢大学
  • Year and Date: 2007-06-13
• [Presentation] ねじれAlexander不変量とその周辺 (2007)
  • Author(s): 合田 洋
  • Organizer: 金沢大学数理学談話会
  • Place of Presentation: 金沢大学
  • Year and Date: 2007-06-13
• [Presentation] Circle-valued Morse maps, Reidemeister torsions, and sutured manifold theory (2007)
  • Author(s): 合田洋
  • Organizer: 日本数学会
  • Place of Presentation: 埼玉大学(特別講演)
  • Year and Date: 2007-03-30
• [Presentation] Lefschetz zeta function and some flows associated with circle valued Morse maps on 3-manifolds (2007)
  • Author(s): Hiroshi Goda
  • Organizer: The 3^<rd> East Asian School of Knots, Links and Related topics
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2007-02-06
• [Presentation] Some estimates of Morse-Novikov number of knots and links (2006)
  • Author(s): 合田洋
  • Organizer: 研究集会「Intelligence of Low Dimensional Topology」
  • Place of Presentation: 広島大学
  • Year and Date: 2006-07-25
• [Presentation] Counting closed orbits and flow lines via Heegaard splittings (2006)
  • Author(s): 合田洋
  • Organizer: トポロジー・幾何セミナー
  • Place of Presentation: 広島大学
  • Year and Date: 2006-07-04
• [Presentation] Counting closed orbits and flow lines via Heegaard splitting (2006)
  • Author(s): 合田洋
  • Organizer: トポロジー金曜セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2006-06-02
• [Presentation] 結び目に対する多重値モース理論について (2006)
  • Author(s): 合田洋
  • Organizer: 低次元トポロジーセミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2006-05-23
• [Presentation] Counting closed orbits and flow lines via Heegaard splittings (2006)
  • Author(s): 合田洋
  • Organizer: トポロジー火曜セミナー
  • Place of Presentation: 東京大学
  • Year and Date: 2006-04-25
• [Book] ねじれAlexander不変量 (2006)
  • Author(s): 北野晃朗, 合田洋, 森藤孝之
  • Total Pages: 183
  • Publisher: 日本数学会
• [Book] ねじれ Alexander 不変量 (2006)
  • Author(s): 北野晃朗, 合田洋, 森藤孝之
  • Total Pages: 183
  • Publisher: 日本数学会