Construction Theorem

 If $f\colon M\to\R^3$ is a tight immersion, then $M$ can be decomposed into two regions, $M^+$ and $M^-$ where: $K\ge 0$ on $M^+$ and $K\le 0$ on $M^-$ $f$ embeds $M^+$ onto the complement in $\dHf$ of a finite number of disjoint planar convex closed disks $D_1,\ldots,D_k$ in $\dHf$ for each $i$, $f^{-1}(\partial D_i)$ is a non-trivial 1-cycle in $M$ (these are called top cycles)