  In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a
neighbourhood basis that consists of nested solid tori $T_i$. We call these
sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a
generalization of the classical notion of genus from knot theory. Here we
introduce the self-geometric index of a toroidal set $K$, which captures how
each torus $T_{i+1}$ winds inside the previous $T_i$. We use this index in
conjunction with the genus to approach the problem of whether a toroidal set
can be realized as an attractor for a flow or a homeomorphism of
$\mathbb{R}^3$. We obtain a complete characterization of those that can be
realized as attractors for flows and exhibit uncountable families of toroidal
sets that cannot be realized as attractors for homeomorphisms.
