Remove specific edge from ST (link-cut) tree
For any edge $(u,v)$ either $u$ is $v$'s parent or vice versa. So you just call CUT for "deeper" vertex.

UPD: To determine which vertex is deeper you can use Access($v$). If after this operation $u$ is a rightmost node of $v$'s left (path)subtree, then $u$ is $v$'s parent and you can call CUT($v$) (or just $\mbox{left}(v) \gets \mbox{Null}$ because Access($v$) was called before). Else you call CUT($u$).