Dynamic planar exact k-nearest neighbors for pathological data
According to TOPP 63, dynamic nearest neighbors can be answered in $O($polylog$)$ time per insert, delete, or query (that is, the case $k=1$). The general case can therefore also be answered in time $O(k$ polylog$)$ per query: to do a query for $k$ points, repeatedly find and delete the nearest neighbor, then once the query is done re-insert all the points you deleted.