Isolated regions of a link projection

A set of regions of a link projection is said to be isolated if any pair of regions in the set share no crossings. The isolate-region number of a link projection is the maximum value of the cardinality for isolated sets of regions of the link projection. In this paper, all the link projections of isolate-region number one are determined. Also, estimations for welded unknotting number and a combinatorial way to find the isolate-region number are discussed, and a formula of the generating function of isolated-region sets is given for the standard projections of [Formula: see text]-torus links.