  We show that any embedding of a planar graph can be encoded succinctly while
efficiently answering a number of topological queries near-optimally. More
precisely, we build on a succinct representation that encodes an embedding of
$m$ edges within $4m$ bits, which is close to the information-theoretic lower
bound of about $3.58m$. With $4m+o(m)$ bits of space, we show how to answer a
number of topological queries relating nodes, edges, and faces, most of them in
any time in $\omega(1)$. Further, we show that with $O(m)$ bits of space we can
solve all those operations in $O(1)$ time.
