Heegaard genus and complexity of fibered knots

Abstract

We prove that if a fibered knot K$K$ with genus greater than 1 in a three‐manifold M$M$ has a sufficiently complicated monodromy, then K$K$ induces a minimal genus Heegaard splitting P$P$ that is unique up to isotopy, and small genus Heegaard splittings of M$M$ are stabilizations of P$P$. We provide a complexity bound in terms of the Heegaard genus of M$M$. We also provide global complexity bounds for fibered knots in the three‐sphere and lens spaces.