How does one make a link-cut tree support finding the maximum (or anything else of the sort) along the edges of a vertex to the root? All implementations I have seen only support finding aggregates of vertices along such paths, and I think that simply having each vertex store the value of the edge from it to its parent doesn't work well when one links vertices which changes the parents of a bunch of vertices.
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The trick is to treat edges as vertices : an edge (u,v) in your forest is represented by two edges in the link-cut tree : (u,e) and (e,v), with the edge information associated to the node e.