Mutation of tilting bundles of tubular type

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