A group $G$ is said to have the property $R_\infty$ if every automorphism
$\varphi \in {\rm Aut}(G)$ has an infinite number of $\varphi$-twisted
conjugacy classes. Recent work of Gon\c{c}alves and Kochloukova uses the
$\Sigma^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\infty}$
property for a certain class of groups, including the generalized Thompson's
groups $F_{n,0}$. In this paper, we make use of the $\Omega^n$ invariants,
analogous to $\Sigma^n$, to show $R_{\infty}$ for certain finitely generated
groups. In particular, we give an alternate and simpler proof of the
$R_{\infty}$ property for BS(1,n). Moreover, we give examples for which the
$\Omega^n$ invariants can be used to determine the $R_{\infty}$ property while
the $\Sigma^n$ invariants techniques cannot.