Applications of Alexander polynomial

• Project/Area Number: 17K05246
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kanazawa University
• Principal Investigator: Kadokami Teruhisa 金沢大学, 機械工学系, 教授 (80382026)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); 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); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 結び目 / アレクサンダー多項式 / Reidemeister torsion / デーン手術 / もろ手性 / 結び目と数論 / 結び目理論 / 低次元トポロジー / Alexander polynomial / Dehn surgery / Seifert fibered space / トポロジー / 幾何学

Outline of Final Research Achievements

The Alexander polynomial is the most classical polynomial invariant for knots, which is always important in Knot Theory and Low dimensional Topology.From the fact that the Alexander polynomial is deeply related with the Reidemeister torsion, which is an invariant for 3-dimensional manifolds, via surgery formula, I have studied the value of the Reidemeister torsion of lens spaces and Seifert manifolds by using the facts from cyclotomic field theory.

Academic Significance and Societal Importance of the Research Achievements

アレクサンダー多項式は学術的に様々な方面に応用できる。私が研究で行った主な応用は、手術理論、絡み目の対称性問題、結び目理論と数論の関連性の理論である。特に数論との関連性からわかるように、今後も他分野との関わりを広げられる可能性を秘めていると確信する。

Research Products

• [Journal Article] Seifert surgery on knots via Reidemeister torsion and Casson-Walker invariant III (2018)
  • Author(s): Teruhisa Kadokami, Noriko Maruyama, Tsuyoshi Sakai
  • Journal Title: Topology and its Applications Volume: 241 Pages: 78-81
  • DOI: 10.1016/j.topol.2018.03.034
  • Peer Reviewed
• [Presentation] Geometric study of virtual knot theory (2019)
  • Author(s): 門上晃久
  • Organizer: 拡大KOOKセミナー2019
• [Presentation] Knot theory in 3-manifold via virtual knot theory (2019)
  • Author(s): Teruhisa Kadokami
  • Organizer: Knots in Tsushima 2019
  • Int'l Joint Research
• [Presentation] Reidemeister torsion の利用法1,2 (2019)
  • Author(s): 門上晃久
  • Organizer: 低次元トポロジー勉強会
• [Presentation] Knot theory in 3-manifold via virtual knot theory (2019)
  • Author(s): 門上晃久
  • Organizer: 結び目の数理II