multicurve

Multicurve is a finite collection of disjoint simple closed curves on a surface, usually taken to be essential and pairwise non-homotopic, and considered up to isotopy. Each curve in the collection is simple (no self-intersections) and closed, and the curves do not intersect each other. In many contexts a multicurve is denoted by C = {c1, ..., ck}.

Geometric viewpoint: In a given hyperbolic structure on the surface, each curve has a unique simple closed geodesic representative; thus the multicurve corresponds to a finite set of pairwise disjoint geodesics. A multicurve is often called maximal if it is not properly contained in any larger multicurve, meaning that adding any additional essential simple closed curve would force an intersection with the existing curves.

Maximal multicurves and decompositions: Cutting the surface along a maximal multicurve decomposes it into a disjoint union of pairs of pants. For a connected orientable surface of genus g with b boundary components, a maximal multicurve has size 3g − 3 + b. Such decompositions are fundamental in the study of Teichmüller theory and the mapping class group.

Variants and uses: A weighted multicurve assigns a positive weight to each component, yielding a measured lamination when combined with a suitable structure. Multicurves underpin pants decompositions, the analysis of Dehn twists in the mapping class group, and the definition of combinatorial objects such as the multicurve complex, whose vertices correspond to isotopy classes of multicurves and encode their disjointness relations.

Examples: On a genus-2 surface, two non-intersecting, non-parallel simple closed curves form a multicurve. On a torus, a single essential simple closed curve is a multicurve.

See also: simple closed curve, pants decomposition, Dehn twist, curve complex, measured lamination.