# A meromorphic extension of the 3D Index

## TL;DR

This paper introduces a meromorphic function associated with ideal triangulations of 3-manifolds with torus boundaries, invariant under Pachner moves, and relates it to the 3D index when a strict angle structure exists.

## Contribution

It constructs a new topological invariant via a meromorphic function that generalizes the 3D index and can be explicitly computed from triangulation data.

## Key findings

- The meromorphic function is invariant under all 2-3 Pachner moves.
- When a strict angle structure exists, the function expands into a Laurent series matching the 3D index.
- Explicit computations are provided for several example manifolds.

## Abstract

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.