Construction of polynomial invariants for knotoids

• Project/Area Number: 17K05255
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Yamaguchi University
• Principal Investigator: Miyazawa Yasuyuki 山口大学, 大学院創成科学研究科, 教授 (60263761)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: knotoid / 結び目 / 結び目理論 / 多項式不変量 / 不変量 / HOMFLY多項式 / enhanced bracket 多項式

Outline of Final Research Achievements

The principal investigator studied "knotoids" which are represented by "open" knot diagrams in a surface and successfully constructed the Jones polynomial, the HOMFLY polynomial and the Kauffman polynomial for knotoids, which correspond to the three famous polynomial invariants in knot theory.

Academic Significance and Societal Importance of the Research Achievements

開発されたknotoidの多項式不変量はknotoidの分類のみならず特質の解明に役立つ。また，結び目理論への応用やその形状と深く関わる他分野，特に，DNA結び目と繋がる生物分野や高分子化合物を対象とする物理・化学分野の諸問題について解決への寄与が期待できる。さらには，その先に続く工学的・農学的分野の応用へと波及し，我々の実生活に好影響を与えるのではないかと想像される・

Research Products

• [Journal Article] A polynomial invariant of Kauffman type for knotoids (2023)
  • Author(s): Yasuyuki Miyazawa
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 32 Issue: 09 Pages: 2350050-2350050
  • DOI: 10.1142/s0218216523500505
  • Peer Reviewed
• [Journal Article] A polynomial invariant for knotoids (2021)
  • Author(s): Yasuyuki Miyazawa
  • Journal Title: Osaka Journal of Mathematics Volume: 58
  • NAID: 120007046102
  • Peer Reviewed / Open Access
• [Journal Article] An oriented knotoid diagram has no characteristic states (2021)
  • Author(s): Yasuyuki Miyazawa
  • Journal Title: Kobe Journal of Mathematics Volume: 38
  • Peer Reviewed
• [Journal Article] An enhanced bracket polynomial for knotoids (2019)
  • Author(s): Yasuyuki Miyazawa
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 28 Issue: 10 Pages: 1950061-1950061
  • DOI: 10.1142/s0218216519500615
  • Peer Reviewed
• [Journal Article] Links with trivial 𝑄-polynomial (2019)
  • Author(s): Yasuyuki Miyazawa
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 71 Issue: 1 Pages: 19-42
  • DOI: 10.2969/jmsj/77167716
  • NAID: 130007557119
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Presentation] KnotoidのKauffman型不変量 (2021)
  • Author(s): 宮澤 康行
  • Organizer: 東京女子大学トポロジーセミナー
  • Invited
• [Presentation] A three variable representation of the HOMFLY polynomial (2020)
  • Author(s): 宮澤 康行
  • Organizer: 2019年度琉球結目セミナー
• [Presentation] A HOMFLY type of invariant for linkoids (2019)
  • Author(s): 宮澤 康行
  • Organizer: 拡大KOOKセミナー2019
• [Presentation] A polynomial invariant for knotoids (2018)
  • Author(s): 宮澤 康行
  • Organizer: 研究集会「拡大KOOKセミナー2018」
• [Presentation] An oriented link diagram has no singular states (2018)
  • Author(s): 宮澤 康行
  • Organizer: 研究集会「2018年度琉球結び目セミナー」
• [Presentation] 多項式不変量に関する最近の個人的未解決問題から (2018)
  • Author(s): 宮澤 康行
  • Organizer: 2017年度琉球結び目セミナー