Translation distance bounds for fibered 3-manifolds with boundary

Abstract

Given $M_φ$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $φ$ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in $M_{φ^n}$ tends to infinity as $n\to\infty$. Additionally, we show that an infinite family of fibered hyperbolic knots has translation distance bounded above by two, satisfying a conjecture by Schleimer which postulates that this behavior should hold for all fibered knots.