A study on the relationship between invariants of homology cylinders, and its applications

• Project/Area Number: 24540068
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Agriculture and Technology
• Principal Investigator: GODA HIROSHI 東京農工大学, 工学(系)研究科(研究院), 教授 (60266913)
• Co-Investigator(Renkei-kenkyūsha): SAKASAI Takuya 東京大学, 大学院数理科学研究科, 准教授 (60451902)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: ホモロジーシリンダー / ファイバー結び目 / 写像類 / モノドロミー / アレキサンダー多項式 / 写像類群

Outline of Final Research Achievements

We focused on 2-bridge fibered knots, whose family is a subset of homologically fibered knots. Then, we obtained a formula of a presentation of its monodromy map as an element in Automorphism of free groups. Using this formula, we gave a formula of the Alexander polynomial. We made a table of elements in Automorphism of free groups, which shows the monodromy maps of homologically fibered 2-bridge knots up to 12 crossings.

Research Products

• [Journal Article] Morse-Novikov theory, Heegaard splittings, and closed orbits of gradient flows (2014)
  • Author(s): Hiroshi Goda, Hiroshi Matsuda, Andrei Pajitnov
  • Journal Title: Algebra i Analiz Volume: 26
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Homology cylinders and sutured manifolds for homologically fibered knots (2013)
  • Author(s): Hiroshi Goda, Takuya Sakasai
  • Journal Title: Tokyo Journal of Mathematics Volume: 36
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots (2012)
  • Author(s): Hiroshi Goda, Chuichiro Hayashi
  • Journal Title: Kobe Journal of Mathematics Volume: 29
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II (2012)
  • Author(s): Hiroshi Goda, Chuichiro Hayashi
  • Journal Title: Saitama Mathematical Journal Volume: 29
  • Peer Reviewed
• [Presentation] ８の字結び目の高次ねじれアレキサンダー多項式 (2015)
  • Author(s): 合田洋
  • Organizer: 第38回伊豆トポロジーセミナー
  • Place of Presentation: 山喜旅館（静岡県伊東市東松原町）
  • Year and Date: 2015-03-17
• [Presentation] 双曲幾何とライデマイスタートーション (2015)
  • Author(s): 合田洋
  • Organizer: 慶應トポロジーセミナー
  • Place of Presentation: 慶応大学（横浜市港北区日吉）
  • Year and Date: 2015-02-02
• [Presentation] ねじれアレキサンダー不変量とファイバー3次元多様体 (2013)
  • Author(s): 合田洋
  • Organizer: 日大トポロジーセミナー
  • Place of Presentation: 東京都世田谷区桜上水３－２５－４０日本大学文理学部
• [Presentation] ファイバー結び目とねじれAlexander不変量 (2013)
  • Author(s): 合田洋
  • Organizer: ENCOUNTER with MATHEMATICS
  • Place of Presentation: 東京都文京区春日１－１３－２７中央大学理工学部
  • Invited
• [Presentation] Monodromies of fibered 2-bridge knots (2013)
  • Author(s): 合田洋
  • Organizer: 群馬大学トポロジーセミナー
  • Place of Presentation: 群馬大学