Unique 3-decomposition and mapping classes of knotted handlebodies

This paper proves a uniqueness result for 2-spheres that split a knotted handlebody in the 3-sphere along three parallel disks. We apply the result to study the symmetry of knotted handlebodies, measured by the mapping class group. In particular, the chirality of 610\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {6_{10}}$$\end{document} in the handlebody-knot table, which was previously unknown, is determined. An infinite family of hyperbolic handlebody-knots with homeomorphic exteriors is also constructed.