Double branched covers of tunnel number one knots

We provide criteria ensuring that a tunnel number one knot K is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} not equivalent to K.