Reformulation and generalization of knot invariants using quandles

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17600
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Sophia University
• Principal Investigator: Oshiro Kanako 上智大学, 理工学部, 准教授 (90609091)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: カンドル / シャドウバイカンドル / local biquandle / カンドル表示 / カンドルコサイクル不変量 / アレクサンダー不変量 / 捩れアレクサンダー不変量

Outline of Final Research Achievements

First, we introduced and studied a new family of knot invariants that includes twisted Alexander invariants and quandle cocycle invariants. We also introduced a generalization of the notion of Fox calculus, which gives a knot invariant obtaind from quandle presentations of knot quandles. This study was given with the cooperation of Atsushi Ishii in University of Tsukuba. Second, we gave an interpretation of knot-theoretic ternary-quasigroup theory (which is a theory corresponding to region colorings of knot diagrams) by using local biquandles. This implies that knot-theoretic ternary-quasigroup theory can be interpreted similary as biqandle theory which is well-known. This study was given with the cooperations of Natsumi Oyamaguchi in Shumei University and Sam Nelson in Claremont McKenna College. Futhermore, we gave a relationship between shadow biquandle theory and knot-theoretic ternary-quasigroup theory.
The obtaiend results was announced in some conferences or in research papers.

Free Research Field

結び目理論

Academic Significance and Societal Importance of the Research Achievements

カンドル代数を用いた様々な結び目不変量の再定式化を考えることで, 不変量の計算の単純化および, 一般化による強力な不変量の構成が期待できる.
本研究では, 捩れアレキサンダー不変量やカンドルコサイクル不変量, knot-theoretic ternary-quasigroup理論のカンドル代数を用いた再定式化を与えた. 特に, 捩れアレキサンダー不変量の再定式化の応用として, 結び目の5-move同値性の判定方法を与えた. このように, 既存結び目不変量の再定式化や一般化によって, 研究の幅や手段が広がり, 今後も新たな具体的計算例や応用例が発見されることが期待される.