The number of Cromwell moves needed for unknotting an arc-presentation of the trivial knot

• Project/Area Number: 22540101
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Japan Women's University
• Principal Investigator: HAYASHI Chuichiro 日本女子大学, 理学部, 教授 (20281321)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 結び目理論 / 自明結び目 / アーク表示 / クロムウェル変形 / 位相幾何学 / 幾何学

Research Abstract

A knot is a circle in the 3-dimensional space R3. Every knot can be placed in a figure in the form of an open-book (a book opened so that every adjacent pair of pages are tangent to each other only in the biding) so that it intersects every page in a single arc). We call such a placement of a knot an arc-presentation. A knot is called trivial if it lies in a plane after being moved continuously. The trivial knot has an arc-presentation with two arcs. An example of infinite sequence of arc-presentations with n arcs of the trivial knot as below is given. They need linearly many exchange moves not changing the number of arcs with respect to n until they admit a merge move decreasing the number of arcs.

Research Products

• [Journal Article] Minimal unknotting sequence of Reidemeister moves containing unmatched RII moves (2012)
  • Author(s): Chuichiro Hayashi, Miwa Hayashi, Minori Sawada and Sayaka Yamada
  • Journal Title: Journal of Knot Theory and its Ramifications Volume: Vol.21, No.10 Pages: 13-13
  • Peer Reviewed
• [Journal Article] Unknotting number and number of Reidemeister moves neededfor unlinking (2012)
  • Author(s): Chuichiro Hayashi, Miwa Hayashi andTahl Nowik
  • Journal Title: Topology and its Applications Volume: Vol.159 Pages: 1467-1474
  • Peer Reviewed
• [Journal Article] On linear n-colorings for knots (2012)
  • Author(s): Chuichiro Hayashi, Miwa Hayashi and Kanako Oshiro
  • Journal Title: Journal of Knot Theory and its Ramifications Volume: Vol.21, No.14 Issue: 14 Pages: 13-13
  • DOI: 10.1142/s0218216512501234
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of1-genus 1-bridge knots (2012)
  • Author(s): Hiroshi Goda and Chuchiro Hayashi
  • Journal Title: Kobe Journal of Mathematics Volume: Vol. 29 Pages: 45-84
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of1-genus 1-bridge knots II (2012)
  • Author(s): Hiroshi Goda and Chuichiro Hayashi
  • Journal Title: SaitamaMathematical Journal Volume: Vol. 29 Pages: 25-54
  • Peer Reviewed
• [Journal Article] Unknotting number and number of Reidemeister moves needed for unlinking. (2012)
  • Author(s): Chuichiro Hayashi
  • Journal Title: Topology and its Applications Volume: 159 Pages: 1467-1474
  • Peer Reviewed
• [Journal Article] Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves. (2012)
  • Author(s): Chuichiro Hayashi
  • Journal Title: Journal of Knot Theory and its Ramifications Volume: 21 Issue: 10 Pages: 1-13
  • DOI: 10.1142/s021821651250099x
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of 1-genus 1-bridge knots. (2012)
  • Author(s): Chuichiro Hayashi
  • Journal Title: Kobe Journal of Mathematics Volume: 29 Pages: 45-84
  • Peer Reviewed
• [Journal Article] Genus two Heegaard splittings of 1-genus 1-bridge knots II (2012)
  • Author(s): Chuichiro Hayashi
  • Journal Title: Saitama Mathematical Journal Volume: 29 Pages: 25-54
  • Peer Reviewed
• [Journal Article] Unknotting number and number Reidemeister moves needed for unlinking (2012)
  • Author(s): Chuichiro Hayashi, Miwa Hayashi, Tahl Nowik
  • Journal Title: Topology and its Applications Volume: 159 Issue: 5 Pages: 1467-1474
  • DOI: 10.1016/j.topol.2012.01.008
  • Peer Reviewed
• [Journal Article] Canonical forms for operation tables of finite connected quandles
  • Author(s): Chuichiro Hayashi
  • Journal Title: Communications in Algebra
  • Peer Reviewed
• [Presentation] 自明結び目のレクタンギュラー・ダイアグラム I (2012)
  • Author(s): 西川友紀(話者)、林忠一郎
  • Organizer: 2012琉球結び目セミナー
  • Place of Presentation: 那覇市ぶんか テンブス館
  • Year and Date: 2012-09-03
• [Presentation] 自明結び目のレクタンギュラー・ダイアグラムII (2012)
  • Author(s): 山田さやか(話者)、林忠一郎
  • Organizer: 2012琉球結び目セミナー
  • Place of Presentation: 那覇市ぶんかテンブス館
  • Year and Date: 2012-09-03
• [Presentation] 自明結び目のレクタンギュラー・ダイアグラムIII (2012)
  • Author(s): 林忠一郎
  • Organizer: 2012琉球結び目セミナー
  • Place of Presentation: 那覇市ぶんかテンブス館
  • Year and Date: 2012-09-03
• [Presentation] 自明結び目のレクタンギュラー・ダイアグラムI (2012)
  • Author(s): 西川友紀
  • Organizer: 2012琉球結び目セミナー
  • Place of Presentation: 那覇市ぶんかテンブス館
• [Presentation] Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves (2011)
  • Author(s): 山田さやか(話者)、澤田実、林忠一郎、 林美和
  • Organizer: 研究集会「結び 目の数学IV」
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-12-27
• [Presentation] CowritheとReidemeister変形の回数-torusknotsへの応用 (2010)
  • Author(s): 林忠一郎(話者)、林美和(話者)
  • Organizer: 結び目の数理セミナーKnottingNagoya
  • Place of Presentation: 名古屋工業大学
  • Year and Date: 2010-06-06
• [Presentation] CowritheとReidemeister変形の回数-torus knotsへの応用 (2010)
  • Author(s): 林忠一郎
  • Organizer: Knotting Nagoya
  • Place of Presentation: 名古屋工業大学
  • Year and Date: 2010-06-06
• [Remarks]
  • URL: http://www2.jwu.ac.jp/kgr/jpn/ResearcherInformation/ResearcherInformation.aspx?KYCD=00006715
• [Remarks] 研究者情報
  • URL: http://www2.jwu.ac.jp/kgr/jpn/ResearcherInformation/ResearcherInformation.aspx?KYCD=00006715
• [Remarks]
  • URL: http://www2.jwu.ac.jp/kgr/jpn/ResearcherInformation/ResearcherInformation.aspx?KYCD=00006715
• [Remarks]
  • URL: http://mcm-www.jwu.ac.jp/~hayashic/index.html