Research on 3-manifolds by using infinite complexes associated with surfaces

• Project/Area Number: 15K04869
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Joetsu University of Education
• Principal Investigator: Saito Toshio 上越教育大学, 大学院学校教育研究科, 教授 (90397670)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: ３次元多様体 / 無限腹帯 / Heegaard分解 / フロースパイン / 結び目 / デーン手術 / タングル / ヘガード分解 / 仮想結び目 / 無限複体 / トンネル数 / 橋指数 / タングル分解

Outline of Final Research Achievements

Investigation on topological and geometrical properties of 3-manifolds is one of the important matters in low dimensional topology. In this research, I focused on surfaces which have strong links with the structure of 3-manifolds. Considering proper criteria with topological and geometrical properties of 3-manifolds, I characterized a part of the structure of 3-manifolds and gave new thought on topological properties of surfaces in 3-manifolds.

Academic Significance and Societal Importance of the Research Achievements

曲面に付随する無限複体を用いることにより，「良い性質」を保証するための条件に対する乖離の度合いを細かく計ることが可能となった。すなわち，一般には無限個のクラスを用意することができ，その条件に纏わる詳細な考察を行うことを可能とした。この点が従来とは大きく異なり，３次元多様体内の曲面に対する幾つかの概念に対する先入観の再考を促すきっかけを与えることができた。

Research Products

• [Int'l Joint Research] The University of South Florida(米国)
• [Int'l Joint Research] 全北大学校(韓国)
• [Int'l Joint Research] University of California, Santa Barbara(米国)
• [Int'l Joint Research] 全北大学校(韓国)
• [Journal Article] Minimal partitions for critical Heegaard splittings (2020)
  • Author(s): Jung Hoon Lee, Toshio Saito
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 29 Issue: 04 Pages: 2050023-2050023
  • DOI: 10.1142/s0218216520500236
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bicompressible surfaces and incompressible surfaces (2019)
  • Author(s): Toshio Saito
  • Journal Title: Bull. Korean Math. Soc. Volume: 56 Pages: 515-520
  • Peer Reviewed
• [Journal Article] High distance tangles and tunnel number of knots (2018)
  • Author(s): Toshio Saito
  • Journal Title: Math. Proc. Cambridge Philos. Soc. Volume: 165 Issue: 3 Pages: 541-548
  • DOI: 10.1017/s0305004117000652
  • Peer Reviewed
• [Journal Article] Cosmetic surgery and the SL(2,C) Casson invariant for two-bridge knots (2018)
  • Author(s): Kazuhiro Ichihara and Toshio Saito
  • Journal Title: Hiroshima Math. J. Volume: 48 Pages: 21-37
  • Peer Reviewed / Open Access
• [Journal Article] Examples of nested essential free tangles (2016)
  • Author(s): Toshio Saito
  • Journal Title: J. Knot Theory Ramifications Volume: 4 Issue: 04 Pages: 1650020-1650020
  • DOI: 10.1142/s0218216516500206
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Combed 3-manifolds as viewed from virtual knot diagrams (2019)
  • Author(s): 斎藤 敏夫
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Tunnel number of knots and generalized tangles (2018)
  • Author(s): 斎藤 敏夫
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Topology and Geometry of Low-dimensional Manifolds (2018)
  • Author(s): 斎藤 敏夫
  • Organizer: Tunnel number of knots and generalized tangles
  • Invited
• [Presentation] An approach to defining Hempel distance of generalized Heegaard splittings (2017)
  • Author(s): 斎藤 敏夫
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] High distance tangles and tunnel number of knots (2016)
  • Author(s): 斎藤 敏夫
  • Organizer: 2016 International Conference of the Honam Mathematical Society
  • Place of Presentation: 全北大学校 (全州，韓国)
  • Int'l Joint Research / Invited
• [Presentation] High distance tangles and tunnel number of knots (2015)
  • Author(s): 斎藤 敏夫
  • Organizer: 拡大KOOKセミナー2015
  • Place of Presentation: 神戸大学 (兵庫県・神戸市)
  • Year and Date: 2015-08-19
• [Book] Lecture notes on generalized Heegaard splittings (2016)
  • Author(s): M. Scharlemann, J. Schultens and T. Saito
  • Total Pages: 138
  • Publisher: World Scientific Publishing Co. Pte. Ltd.
• [Remarks] 斎藤 敏夫 (さいとう としお)
  • URL: http://www.juen.ac.jp/math/saito/