  Immersion (i.e., local embedding) is relevant to the physics of topologically
ordered phases through entanglement bootstrap. An annulus can immerse 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 appendices, we discuss related problems in broader contexts.
