  In polymer science, cross-linking of polymer chains yields a substantially
modified system compared to the one-dimensional constituent chains, due to the
increase of dimensionality and effective seeding by defects (cross-linking
sites). Inspired by this concept, we analyze topological features of a unit
cell of a generalized topological mesh comprised of several one-dimensional
Su-Schrieffer-Heeger (SSH) lattices cross-linked via a single site. The
coupling site functions as a defect with protected states in the trivial regime
and also induces edges inside the bulk with protected localized states centered
around it in the topological regime. When more than two lattices are coupled by
the defect, namely, a graph-dimensionality larger than one, the crossed chains
support two types of localized eigenstates around the defect. One type is
highly controllable by modifying the cross-linking strength, enabling broad
tuning of eigenenergies from being submerged in the bulk band to becoming
highly isolated and protected. We show that these unique features can be
explained by an equivalence of the aforementioned system to an SSH chain
coupled nonreciprocally to an external reservoir, yielding a unique
pseudospectrum for both the bulk and localized states, with spatially symmetric
eigenstates. Applying non-Hermiticity by adding gain and loss to alternating
sites, relevant for example to a possible realization of topological
coupled-laser fabric, we observe an abrupt transition of the topologically
protected mid-gap state from anti-localized to localized near the defect. By
changing the gain-loss parameters, we observe a cascaded spatial symmetry
breaking of the supported states at exceptional points where parity-time
symmetry is broken, for both the localized and the bulk states, exhibiting
various novel phases.
