High distance bridge surfaces

Abstract

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 $\left(M,L\right)$ of genus $g$ that intersects $L$ in $2b$ points and has distance at least $n$. More generally, given two possibly disconnected surfaces $S$ and $S^{\prime }$, 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 $S\cup S^{\prime }$ such that $M$ contains a $c$–component tangle $T$ with a bridge surface $\Sigma$ of genus $g$ that separates $\partial M$ into $S$ and $S^{\prime }$, $|T\cap \Sigma |=2b$ and $T$ intersects $S$ and $S^{\prime }$ exactly in their marked points, and $\Sigma$ has distance at least $n$.