High Distance Bridge Surfaces

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S', each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S \cup S' such that M contains a c-component tangle T with a bridge surface Sigma of genus g that separates the boundary of M into S and S', |T \cap Sigma|=2b and T intersects S and S' exactly in their marked points, and Sigma has distance at least n.