Immersed figure-8 annuli and anyons

Immersion (i.e., local embedding) is relevant to the physics of topologically ordered phases through the entanglement bootstrap. An annulus can be immersed in a disk or a sphere as a "figure-8", which cannot be smoothly deformed to an embedded annulus. We investigate a simple problem: is there an Abelian state on the immersed figure-8 annulus, locally indistinguishable from the ground state of the background physical system? We show that if the answer is affirmative, a strong sense of isomorphism must hold: two homeomorphic immersed regions must have isomorphic information convex sets, even if they cannot smoothly deform to each other on the background physical system. We explain why to care about strong isomorphism in physical systems with anyons and give proof in the context of Abelian anyon theory. We further discuss a connection between immersed annuli and anyon transportation in the presence of topological defects. In the appendices, we discuss related problems in broader contexts.