Using the locally compact abelian group \(\mathbb {T}\times \mathbb {Z}\), we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2–3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a strict angle structure, our meromorphic function can be expanded into a Laurent power series whose coefficients are formal power series in q with integer coefficients that coincide with the 3D index of (Dimofte et al. in Adv Theor Math Phys 17(5):975–1076, 2013). Our meromorphic function can be computed explicitly from the matrix of the gluing equations of a triangulation, and we illustrate this with several examples.

中文翻译：

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3D索引的亚纯扩展

使用局部紧致的阿贝尔群\（\ mathbb {T} \ times \ mathbb {Z} \），我们将亚纯函数分配给具有圆环边界分量的3形流形的每个理想三角剖分。该函数在所有2–3个Pachner移动下都是不变的，因此是基础流形的拓扑不变性。如果理想的三角剖分具有严格的角度结构，则我们的亚纯函数可以扩展为Laurent幂级数，其系数为q的形式幂级数，其整数系数与（Dimofte等人的Adv Theor Math Phys 17 （5）：975-1076，2013年）。我们的亚纯函数可以从三角剖分的胶合方程矩阵中显式计算，并通过几个示例进行说明。