Handlebody-knots and augmented Alexander invariants

• Project/Area Number: 18K03292
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: University of Tsukuba
• Principal Investigator: Ishii Atsushi 筑波大学, 数理物質系, 准教授 (00531451)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 結び目理論

Outline of Final Research Achievements

A handlebody-knot is a handlebody embedded in the 3-sphere. The notion of a handlebody-knot is a natural generalization of that of a knot, since a genus one handlebody-knot corresponds to a usual knot. The results of this study are as follows: We defined an MCQ twisted Alexander ideal of handlebody-knots. We present a method to obtain an MCQ Alexander pair from a quandle Alexander pair, where an MCQ Alexander pair is a pair of maps that is used to define an MCQ twisted Alexander ideal.

Academic Significance and Societal Importance of the Research Achievements

結び目理論はひもという素朴なものを媒体に、空間の形やＤＮＡ、暗号理論など様々な分野と関わりを持っています。ハンドル体結び目理論は結び目理論の一分野で、空間に埋め込まれたハンドル体を研究対象にします。結び目は粒子の運動の軌跡と捉えることができますが、ハンドル体結び目はこれらの粒子に分裂・合体を許したものに対応します。二つのハンドル体結び目を判別するために不変量が必要になりますが、本研究ではＭＣＱアレクサンダーイデアルと呼ばれる不変量を構築しました。

Research Products

• [Journal Article] Cocycles of G-Alexander biquandles and G-Alexander multiple conjugation biquandles (2021)
  • Author(s): Ishii Atsushi、Iwakiri Masahide、Kamada Seiichi、Kim Jieon、Matsuzaki Shosaku、Oshiro Kanako
  • Journal Title: Topology and its Applications Volume: 301 Pages: 107512-107512
  • DOI: 10.1016/j.topol.2020.107512
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Row relations of twisted Alexander matrices and shadow quandle 2-cocycles (2021)
  • Author(s): Ishii Atsushi、Oshiro Kanako
  • Journal Title: Topology and its Applications Volume: 301 Pages: 107513-107513
  • DOI: 10.1016/j.topol.2020.107513
  • Peer Reviewed
• [Journal Article] A multiple group rack and oriented spatial surfaces (2020)
  • Author(s): Atsushi Ishii, Shosaku Matsuzaki and Tomo Murao
  • Journal Title: J. Knot Theory Ramifications Volume: 29 Issue: 07 Pages: 2050046-2050046
  • DOI: 10.1142/s0218216520500467
  • Peer Reviewed
• [Journal Article] Biquandle (co)homology and handlebody-links (2018)
  • Author(s): Atsushi Ishii, Masahide Iwakiri, Seiichi Kamada, Jieon Kim, Shosaku Matsuzaki, Kanako Oshiro
  • Journal Title: J. Knot Theory Ramifications Volume: 27 Issue: 11 Pages: 1843011-1843011
  • DOI: 10.1142/s0218216518430113
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] On a quandle derivative (2020)
  • Author(s): 石井敦
  • Organizer: カンドルと対称空間
  • Invited
• [Presentation] Generalized quandle cocycle invariants and shadow quandle cocycle invariants (2019)
  • Author(s): Atsushi Ishii
  • Organizer: The Third Pan Pacific International Conference on Topology and Applications
  • Int'l Joint Research
• [Presentation] Fox derivatives for quandles (2019)
  • Author(s): 石井敦
  • Organizer: 日本数学会年会