On developments and applications of Heegaard theory

• Project/Area Number: 21540082
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nara Women's University
• Principal Investigator: KOBAYASHI Tsuyoshi 奈良女子大学, 自然科学系, 教授 (00186751)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 三次元多様体 / Heegaard 分解 / 距離 / ミウラ折り / Heegaard分解 / 相似構造 / Hempel distance / 安定写像 / graphic / 三次元双曲多様体 / 結び目 / 橋指数 / Heeaard分解 / Hemel distance

Research Abstract

In this research we first proved the following two results for Heegaard splitting of 3-manifold which is a generalization of 3-dimensional space: (1)Haken type theorem for essential laminations, and (2) existence of Heegaard splitting of distance n for each positive integer n. Then as an application of low dimensional topology, we gave: (3) a construction of generalized Miura map folding by using similarity structure on 2-dimensional torus.

Research Products

• [Journal Article] A linear bound on the tetrahedral number of manifolds of bounded volume (after Jorgensen and Thurston) (2011)
  • Author(s): T.Kobayashi, and Y.Rieck
  • Journal Title: Contemporary Mathematics Volume: 560 Pages: 27-42
  • Peer Reviewed
• [Journal Article] A design for pseudo-Anosov braids using hypotrochoid curves (2010)
  • Author(s): T.Kobayashi, and S.Umeda
  • Journal Title: Topology and its Appl Volume: 157 Pages: 280-289
  • Peer Reviewed
• [Journal Article] Destabilizing Heegaard splittings of knot exteriors (2010)
  • Author(s): T.Kobayashi, and T.Saito
  • Journal Title: Topology and its Appl Volume: 157 Pages: 202-212
  • Peer Reviewed
• [Journal Article] Destabilizing Heegaard splittings of knot exteriors (2010)
  • Author(s): T.Kobayashi, T.Saito
  • Journal Title: Topology and its Appl. Volume: 157 Pages: 201-212
  • Peer Reviewed
• [Journal Article] A design for gseudo-Anosov braids using hypotrochoid curves (2010)
  • Author(s): T.Kobayashi, S.Umeda
  • Journal Title: Topology and its Appl. Volume: 157 Pages: 280-289
  • Peer Reviewed
• [Journal Article] Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings (2009)
  • Author(s): T.Kobayashi, and Y.Rieck
  • Journal Title: Communications in Analysis and Geometry Volume: 17 Pages: 637-649
  • Peer Reviewed
• [Journal Article] Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings (2009)
  • Author(s): T.Kobayashi, Y.Rieck
  • Journal Title: Communications in Analysis and Geometry 17 Pages: 637-649
  • Peer Reviewed
• [Journal Article] Knots with g(E(K))=2 and g(E(K # K # K))=6 and Morimoto's Conjecture (2009)
  • Author(s): T.Kobayashi, Y.Rieck
  • Journal Title: Topology and its Appl. 156 Pages: 1114-1117
  • Peer Reviewed
• [Presentation] 2 次元トーラスの相似構造を用いた一般化されたミウラ折りの構成 (2013)
  • Author(s): 入井 美紀,小林 毅
  • Organizer: Spring Workshop 2013 on Low-Dimensional Topology and its Ramifications
  • Place of Presentation: 名城大学名駅サテライトキャンパス
  • Year and Date: 2013-03-03
• [Presentation] 2次元トーラスの相似構造を用いた一般化されたミウラ折りの構成 (2013)
  • Author(s): 小林毅
  • Organizer: Spring Workshop 2013 on Low-Dimensional Topology and its Ramifications
  • Place of Presentation: 名城大学名駅サテライトキャンパス
• [Presentation] On the growth rate of tunnel numbers of knots (2011)
  • Author(s): 小林毅
  • Organizer: Workshop on Low Dimensional Topology in Shanghai and Suzhou
  • Place of Presentation: Shanghai and Suzhou
  • Year and Date: 2011-05-17
• [Presentation] On the growth rate of tunnel numbers of knots (2011)
  • Author(s): T.Kobayashi
  • Organizer: Workshop on Low Dimensional Topology in Shanghai and Suzhou
  • Place of Presentation: Suzhou University, School of Mathematical Sciences(招待講演)
  • Year and Date: 2011-05-17
• [Presentation] Toward Haken type theorems for essential laminations in 3-manifolds: Proposal of fundamental settings and applications (2010)
  • Author(s): 小林毅,村井紘子
  • Organizer: E-KOOKセミナー2010
  • Place of Presentation: 大阪市立大学学術情報センター 10 階(杉本キャンパス)
  • Year and Date: 2010-08-28
• [Presentation] Toward Haken type theorems for essential laminations in 3-manifolds : Proposal of fundamental settings and applications (2010)
  • Author(s): 村井紘子、小林毅
  • Organizer: E-KOOKセミナー2010
  • Place of Presentation: 大阪市立大学学術情報センター
  • Year and Date: 2010-08-28
• [Presentation] On thin presentations of 3-manifolds and links (2009)
  • Author(s): 小林毅
  • Organizer: Category Theory, Computer Science, and Topology
  • Place of Presentation: 信州大学理学部
  • Year and Date: 2009-12-18
• [Presentation] On thin presentations of 3-manifolds and links (2009)
  • Author(s): 小林毅
  • Organizer: Category Theory, Computer Science, and Topology
  • Place of Presentation: 信州大学
  • Year and Date: 2009-10-18
• [Presentation] Realization problems of distances of Heegaard splittings (2009)
  • Author(s): 小林毅
  • Organizer: Workshop: Simplicial Complexes Arising in Low-Dimensional Topology
  • Place of Presentation: 東京工業大学
  • Year and Date: 2009-07-03
• [Presentation] Realization problems of distances of Heegaard splittings (2009)
  • Author(s): 小林毅
  • Organizer: Workshop : Simplicial Complexes Arising in Low-Dimensional Topology
  • Place of Presentation: 東京工業大学
  • Year and Date: 2009-07-03
• [Remarks]
  • URL: http://www.math.nara-wu.ac.jp/personal/tsuyoshi/index-j.htm
• [Remarks]
  • URL: http://www.math.nara-wu.ac.jp/personal/tsuyoshi/index-j.htm
• [Remarks]
  • URL: http://www.math.nara-wu.ac.jp/personal/tsuyoshi/index-j.htm
• [Remarks]
  • URL: http://www.math.nara-wu.ac.jp/personal/tsuyoshi/index-j.htm