Mutation of tilting bundles of tubular type

Abstract Let X be a weighted projective line of tubular type and coh X the category of coherent sheaves on X . The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles in coh X is connected. This yields an alternative proof for the connectedness of the tilting graph of coh X . Our approach leads to the investigation of the change of slopes of a tilting sheaf in coh X under (co-)APR mutations, which may be of independent interest.