  Tutte showed that a graph $G$ is planar if and only if the conflict graph
associated to every cycle of $G$ is bipartite. We define a (not necessarily
unique) signed conflict graph associated to a maximally planar subgraph of a
nonplanar graph such that if $G$ has a flat embedding, every possible conflict
graph associated to every maximally planar subgraph of $G$ is balanced. In
doing this, we show that for every graph $G$ with flat embedding, and a planar
subgraph $P$ of $G$, $P$ lies on a sphere that intersects $G$ only in $P$. We
conjecture that $G$ is intrinsically linked if and only if every maximal planar
subgraph of $G$ has every possible conflict graph unbalanced.
