  Some interfaces between two different topologically ordered systems can be
gapped. In earlier work it has been shown that such gapped interfaces can
themselves be effective one dimensional topological systems that possess
localized topological modes in open boundary geometries. Here we focus on how
this occurs in the context of an interface between two, single-component
Laughlin states of opposite chirality, and with filling fractions $\nu_1=1/p$
and $\nu_2=1/pn^2$. While one type of interface in such systems has been
previously studied, we show that allowing for edge reconstruction effects opens
up a wide variety of possible gapped interfaces depending on the number of
divisors of $n.$ We apply a complementary description of the $\nu_2=1/pn^2$
system in terms of Laughlin states coupled to a discrete gauge $\mathbb{Z}_n$
field. This enables us to identify possible interfaces to the $\nu_1$ system
based on complete or partial confinement of this gauge field. We determine the
tunneling properties, ground state degeneracy, and the nature of the
non-Abelian zero modes of each interface in order to physically distinguish
them.
