Integral region choice problems on link diagrams

Abstract

Shimizu introduced a region crossing change unknotting operation for knot diagrams. As extensions, two integral region choice problems were proposed and the existences of solutions of the problems were shown for all non-trivial knot diagrams by Ahara and Suzuki, and Harada. We relate both integral region choice problems with an Alexander numbering for regions of a link diagram, and give alternative proofs of the existences of solutions for knot diagrams. We also discuss the problems on link diagrams. For each of the problems on the diagram of a two-component link, we give a necessary and sufficient condition that there exists a solution.