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.
Categories: math.RT