Abstract
A simple closed curve C in the boundary of a handlebody V is a primitive curve if there exists a disk D in V such that C intersects @D transversely in a single point. Such a disk D is called a dual disk of C, and (C;D) is called a dual pair. Two primitive curves C and C′ in are said to be separated if there exist dual disks D and D′ of C and C′ respectively such that C [ D and C′ [ D′ are disjoint. We show that for any primitive curves C and C′ in the boundary of a genus-g ( 2) handlebody, there exists a sequence C = C1;C2; : : : ;Cn = C′ of primitive curves such that Ci and Ci+1 are separated for each i 2 f1; 2; : : : ; n 1g. As a consequence, the primitive curve complex and the separating disk complex of a genus-g handlebody are connected for every g 2.
Definitions
We explain what a primitive curve is. We define the notion of a dual disc to a primitive curve. dual discs are always non-separating.
We define a curve complex as a complex form of the Kurvengraph. We define the primitive curve complex as the complex whose vertices are the isotopy classes of vertices.
Next we generalize the primitive curves to primitive discs.