A meromorphic extension of the 3D Index

Sunday, 25 June, 2017

Published in: 

arXiv:1706.08132

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.

Author(s): 

Stavros Garoufalidis
Rinat Kashaev