Research on invariants of hyperbolic manifolds centered around twisted Alexander polynomials

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03487
• 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: Tokyo University of Agriculture and Technology
• Principal Investigator: Goda Hiroshi 東京農工大学, 工学(系)研究科(研究院), 教授 (60266913)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: knot / Alexander polynomial / 3-manifold / zeta function

Outline of Final Research Achievements

We have succeeded in presenting the twisted Alexander polynomials using the zeta function for the matrix-weighted graph obtained from a knot diagram. We investigated the relationship between the geometric structure of a cone manifold and the twisted Alexander polynomials, and described the change of the twisted Alexander polynomials according to the deformation of the cone structure of a hyperbolic manifold. Furthermore, we obtained the conditions for the coefficients of the twisted Alexander polynomials of knots in the mirror image relationship.

Free Research Field

低次元トポロジー

Academic Significance and Societal Importance of the Research Achievements

先行研究によりねじれアレキサンダー多項式と結び目の双曲体積の関係が知られているが、それを起点に色付きジョーンズ多項式と結び目の双曲体積の関係を調査するための糸口としてねじれアレキサンダー多項式のゼータ関数を用いた表示を得ることができた．また双曲多様体の錐構造とねじれアレキサンダー多項式との記述もある程度できた．結び目およびその補空間の幾何学的構造を結び目不変量を使って解明、記述することに貢献することができた．