Torsion invariants for hyperbolic manifolds

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04868
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• 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)
• Research Collaborator: KITANO TERUAKI 創価大学, 理工学部, 教授 (90272658) ; MORIFUJI TAKAYUKI 慶應義塾大学, 経済学部, 教授 (90334466) ; YAMAGUCHI YOSHIKAZU 秋田大学, 教育文化学部, 准教授 (30534044)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: ねじれアレキサンダー多項式 / 結び目 / 体積

Outline of Final Research Achievements

A hyperbolic knot group G has a representation to PSL(2,C), which is called holonomy representation. We focused on a representation G ->SL(n,C) obtained from the holonomy representation by the extension, and then we studied the twisted Alexander polynomials associated with the representation.

We have calculated the twisted Alexander polynomials for the figure eight knot and the Whitehead link. Using the results, we have obtained a formula of the volume of a hyperbolic link complement using the twisted Alexander polynomial.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

体積は幾何学において極めて重要な基本概念であり，アレキサンダー多項式は結び目理論において最も重要だと考えられる結び目の多項式不変量である．ねじれアレキサンダー多項式はアレキサンダー多項式を精密な形に拡張したものであり，本研究にて得られたねじれアレキサンダー多項式を用いて双曲結び目補空間の体積を記述する明示公式は結び目研究の歴史に残る公式である．