The 3D-index and normal surfaces

TL;DR

This paper explains how the 3D-index, an invariant of 3-manifolds, can be interpreted as a generating series of normal surfaces, linking quantum invariants with classical topology.

Contribution

It establishes a connection between the 3D-index and normal surface theory, providing a new topological interpretation of the invariant.

Findings

The 3D-index is a generating series of normal surfaces.

This connection bridges quantum invariants and classical topology.

It advances the understanding of 3-manifold invariants.

Abstract

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.