Heegaard genus and complexity of fibered knots

TL;DR

This paper investigates the relationship between fibered knots with complex monodromies and their induced minimal genus Heegaard splittings, establishing uniqueness and stabilization properties, along with complexity bounds in various 3-manifolds.

Contribution

It proves that highly complicated fibered knots induce unique minimal genus Heegaard splittings and characterizes the stabilization of small genus splittings, providing new complexity bounds.

Findings

Unique minimal genus Heegaard splitting induced by complex fibered knots

Small genus splittings are stabilizations of the minimal one

Global complexity bounds for fibered knots in S^3 and lens spaces

Abstract

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