  Topological quantum field theories (TQFT) encode properties of quantum states
in the topological features of abstract manifolds. One can use the topological
avatars of quantum states to develop intuition about different concepts and
phenomena of quantum mechanics. In this paper we focus on the class of simplest
topologies provided by a specific TQFT and investigate what the corresponding
states teach us about entanglement. These ``planar connectome" states are
defined by graphs of simplest topology for a given adjacency matrix. In the
case of bipartite systems the connectomes classify different types of
entanglement matching the classification of stochastic local operations and
classical communication (SLOCC). The topological realization makes explicit the
nature of entanglement as a resource and makes apparent a number of its
properties, including monogamy and characteristic inequalities for the
entanglement entropy. It also provides tools and hints to engineer new measures
of entanglement and other applications. Here the approach is used to construct
purely topological versions of the dense coding and quantum teleportation
protocols, giving diagrammatic interpretation of the role of entanglement in
quantum computation and communication. Finally, the topological concepts of
entanglement and quantum teleportation are employed in a simple model of
information retrieval from a causally disconnected region, similar to the
interior of an evaporating black hole.
