Integral region choice problems on link diagrams

TL;DR

This paper explores integral region choice problems in link diagrams, relating them to Alexander numberings, providing alternative proofs for knots, and establishing conditions for solutions in two-component links.

Contribution

It connects integral region choice problems with Alexander numberings and offers new proofs and conditions for solutions in link diagrams.

Findings

Solutions exist for all non-trivial knot diagrams.

Necessary and sufficient conditions are provided for two-component links.

Alternative proofs for the existence of solutions are presented.

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.