Neutral surfaces, along which most of the mixing in the ocean occurs, are notoriously difficult objects: they do not exist as well-defined surfaces, and as such can only be approximated. In a hypothetical ocean where neutral surfaces are well-defined, the in-situ density on the surface is a multivalued function of the pressure on the surface, $\underset{\sim}{p}$. The surface is decomposed into geographic regions where there is one connected pressure contour per pressure value, making this function single-valued in each region. The regions are represented by arcs of the Reeb graph of $\underset{\sim}{p}$. The regions meet at saddles of $\underset{\sim}{p}$ which are represented by internal nodes of the Reeb graph. Leaf nodes represent extrema of $\underset{\sim}{p}$. Cycles in the Reeb graph are created by islands and other holes in the neutral surface. This topological theory of neutral surfaces is used to create a new class of approximately neutral surfaces in the real ocean, called topobaric surfaces, which are very close to neutral and fast to compute. Topobaric surfaces are the topologically correct extension of orthobaric density surfaces to be geographically dependent, which is fundamental to neutral surfaces. Also considered is the possibility that helical neutral trajectories might have a larger pitch around islands than in the open ocean.

22 pages, 7 figures