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.