  The twisted index of 3d $\mathcal{N}=2$ gauge theories on $S^1 \times \Sigma$
has an algebro-geometric interpretation as the Witten index of an effective
supersymmetric quantum mechanics. In this paper, we consider the contributions
to the supersymmetric quantum mechanics from topological saddle points in
supersymmetric localisation of abelian gauge theories. Topological saddles are
configurations where the matter fields vanish and the gauge symmetry is
unbroken, which exist for non-vanishing effective Chern-Simons levels. We
compute the contributions to the twisted index from both topological and
vortex-like saddles points and show that their combination recovers the
Jeffrey-Kirwan residue prescription for the twisted index and its
wall-crossing.
