Research of relation among quantum invariant and number theoretic invariants and modular forms

量子不变量与数论不变量及模形式关系的研究

• 批准号: 17540067
• 负责人: TAKATA Toshie
• 金额: $ 1.54万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540067/
• 关键词: 3-manifold; knot; quantum invariant; modular form; theta function

Applying to our formula of the colored Jones polynomial for 2-bridge knots Lawrence and Ron's way to compute SU(2)Witten--Reshetikhin--Turaev (WRT) invariant of homology 3-sphere obtained by surgery of a knot from the colored Jones polynomial of the knot, we conjectured some number theoretical properties of the Ohtsuki invariants. Later on, using some results about the colored Jones polynomial given by K.Habiro, we showed the conjecture. Furthermore, using the values of the Ohtsuki invariants for Seifert 3-manifolds given by Hikami, we identified the set of the LMO invariant up to degree 6 for integral homology 3-spheres, and characterized the set of the Ohtsuki invariants up to degree 6.Lawrence and Zagier pointed out that WRT invariant of the Poincare homology sphere is related to the vector modular form with half-integral weight. Namely the WRT invariant coincides with a certain limit of the Eichler integral of the modular forms. As a generalization of this result, we have computed the WRT invariant for the spherical Seifert manifold with three exceptional fibers, and we have found that it coincides with a limiting value of Eicher integral of vector modular form with half-integral weight. By use of the modular property, we have obtained an exact asymptotic expansion of the WRT invariant, and have given an interpretation of topological invariants such as the Ohtsuki series and the Chern--Simons invariant from the viewpoint of modular forms. Furthermore, we have pointed out that the vector modular form is related to the fundamental group of manifolds.

应用我们的 2 桥结彩色琼斯多项式公式，劳伦斯和罗恩的方法来计算 SU(2)Witten--Reshetikhin--Turaev (WRT) 同调 3-球面不变量，通过手术从彩色琼斯结获得通过结的多项式，我们猜想了 Ohtsuki 不变量的一些数论性质。随后，利用K.Habiro给出的有色琼斯多项式的一些结果，我们证明了这个猜想。此外，利用 Hikami 给出的 Seifert 3 流形的 Ohtsuki 不变量值，我们识别了积分同源 3-球体的 LMO 不变量的 6 级集合，并表征了 Ohtsuki 不变量的 6 级集合。 Lawrence和Zagier指出Poincare同调球的WRT不变量与半积分权向量模形式有关。即WRT不变量与模形式的艾希勒积分的一定极限一致。作为这个结果的推广，我们计算了具有三个特殊纤维的球形 Seifert 流形的 WRT 不变量，并且我们发现它与具有半积分权重的向量模形式的 Eicher 积分的极限值一致。利用模性质，我们得到了WRT不变量的精确渐近展开，并从模形式的角度对大月级数、陈-西蒙斯不变量等拓扑不变量给出了解释。此外，我们还指出向量模形式与流形的基本群有关。

项目成果

Mock (false) theta functions as quantum invariants

• DOI: 10.1070/rd2005v010n04abeh000328
• 发表时间: 2005-06
• 期刊: Regular & Chaotic Dynamics
• 影响因子: 1.4
• 作者: K. Hikami

Transformation Formula for the "2nd" Order Mock Theta Function

“二阶”模拟 Theta 函数的变换公式

• 发表时间: 2006
• 期刊: Letters in Mathematical Physics 75
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘
• 通讯作者: 樋上 和弘

A Formula for the Colored Jones Polynomial of 2-Bridge Knots

2 桥结的彩色琼斯多项式的公式

• 发表时间: 2008
• 期刊: Kyungpook Mathematical Journal Vol.48, No.2
• 作者: 高田敏恵

Quantum Invariants, Modular Forms, and Lattice Points II

量子不变量、模形式和格点 II

• 发表时间: 2006
• 期刊: J.Math.Phys. 47
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘;樋上 和弘;K.Hikami;K.Hikami;K.Hikami
• 通讯作者: K.Hikami

On the Topological Structure of Fractal Tilings Generated by Quadratic Number Systems

二次数系统生成的分形瓦片的拓扑结构

• 发表时间: 2005
• 期刊: Computers and Mathematics with Applications 49
• 作者: K. Mine;K. Sakai and M. Yaguchi;樋上 和弘;樋上 和弘;樋上 和弘;K.Hikami;K.Hikami;K.Hikami;樋上 和弘;樋上 和弘;秋山 茂樹
• 通讯作者: 秋山 茂樹