Abstract
When trying to do surgery on contact 3-manifolds we need to understand contact structures in neighborhoods of embedded surfaces. As we already pointed out in Chapter 4, for a given surface Σ ⊂ (Y, ξ) the characteristic foliation F Σ determines the contact structure near Σ. But it is not easy to describe or relate characteristic foliations. It turns out that the same information can be captured by certain configurations of curves on the surface at hand once the surface is in a special position with respect to the contact structure. This theory has been developed and fruitfully applied by Giroux and Honda in various circumstances in 3-dimensional contact geometry. For the sake of completeness, in this Chapter we recall the fundamental definitions and results regarding convex surfaces and dividing sets. These statements will be used in our study of contact Dehn surgery in Chapter 11. For a more detailed introduction to the subject see [43, 76].
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