Access Is Everything

It compresses the virtual edges above $x$ into a single parent pointer, at $O(\alpha(n))$ amortized. link, cut, findroot and split are thin wrappers around it, so they inherit that bound.

It makes the root-to-$x$ path a single preferred path held in one splay tree, at $O(\log n)$ amortized. link, cut, findroot and split are thin wrappers around it, so they inherit that bound.

It rebuilds the heavy light decomposition of the current tree into one balanced structure, at $O(\log^2 n)$ amortized. link, cut, findroot and split are thin wrappers around it, so they inherit that bound.

It rotates $x$ to the root of the represented tree, held in one splay tree, at $O(\log n)$ amortized. link, cut, findroot and makeroot become free once it has run, so they inherit no further cost.