By Emily Hutchinson illustrated by Valeria Petrone

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10 Let α and β be two essential simple closed curves in a surface S. Then α is isotopic to β if and only if α is homotopic to β. 36 CHAPTER 1 Proof. One direction is vacuous, since an isotopy is a homotopy. So suppose that α is homotopic to β. We immediately have that i(α, β) = 0. By performing an isotopy of α, we may assume that α is transverse to β. If α and β are not disjoint then by the bigon criterion they form a bigon. A bigon prescribes an isotopy that reduces intersection. Thus, we may remove bigons one by one by isotopy until α and β are disjoint.

The minimum of {k, g − k} is called the genus of the separating simple closed curve. By the above, the genus of a curve determines and is determined by its topological type. Note that there are ⌊ 2g ⌋ topological types of essential separating simple closed curves in a closed surface. The uninitiated may have trouble visualizing separating simple closed curves that are not the “obvious” ones. 5, and we encourage the reader to draw even more complicated separating simple closed curves. 5 Some nonobvious separating simple closed curves.

A multicurve in S is the union of a finite collection of disjoint simple closed curves in S. The notion of intersection number extends directly to multicurves. 7) gives a version of the bigon criterion for multicurves: two multicurves are in minimal position if and only if no two component curves form a bigon. g. the geodesic) so that each pair of curves is in minimal position. 5 H OMOTOPY VERSUS ISOTOPY FOR SIMPLE CLOSED CURVES Two simple closed curves α and β are isotopic if there is a homotopy H : S 1 × [0, 1] → S from α to β, with the property that the closed curve H(S 1 × {t}) is simple for each t ∈ [0, 1].