Multiple bridge surfaces restrict knot distance

Abstract

Suppose $M$ is a closed irreducible orientable 3–manifold, $K$ is a knot in $M$, $P$ and $Q$ are bridge surfaces for $K$ and $K$ is not removable with respect to $Q$. We show that either $Q$ is equivalent to $P$ or $d\left(K,P\right)\le 2-\chi \left(Q-K\right)$. If $K$ is not a 2–bridge knot, then the result holds even if $K$ is removable with respect to $Q$. As a corollary we show that if a knot in $S^3$ has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.