Using the locally compact abelian group $\BT \times \BZ$, 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
\cite{DGG2}. Our meromorphic function can be computed explicitly from the
matrix of the gluing equations of a triangulation, and we illustrate this with
several examples.Comment: 34 pages, 4 figure

Garoufalidis, Stavros

Kashaev, Rinat

English

arXiv

Using the locally compact abelian group $\BT \times \BZ$, 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 seriesin $q$ 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.<br

A meromorphic extension of the 3D Index

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