Tangent circle bundles admit positive open book decompositions along arbitrary links

In the present paper we generalize the divide lying in the unit disk, introduced by AʹCampo, to compact, oriented, smooth surfaces, and prove a fibration theorem for generalized divides. As a consequence, we will show that, for any link L in the tangent circle bundle Y to the compact surface, there exists an additional knot K such that the link L∪K is the binding of a “positive” open book decomposition of Y.