# Translation distance bounds for fibered 3-manifolds with boundary

## TL;DR

This paper establishes bounds on the translation distance of monodromies in fibered 3-manifolds with boundary, linking it to surface complexity, and confirms a conjecture for a class of fibered hyperbolic knots.

## Contribution

It provides new bounds on translation distance in fibered 3-manifolds and verifies a conjecture for certain fibered hyperbolic knots.

## Key findings

- Translation distance is bounded by the complexity of essential surfaces.
- Complexity of surfaces with non-zero slope tends to infinity with n.
- An infinite family of fibered hyperbolic knots has translation distance at most two.

## Abstract

Given $M_\varphi$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $\varphi$ 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 $M_{\varphi^n}$ 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.