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.
Acknowledgments
The author would like to thank Megumi Hashizume for giving valuable advice and a lot of information about a region crossing change. The author also would like to thank Yasuyoshi Yonezawa for his suggestion and Shingo Harada for his works on an alternating integral region choice problem. She is also grateful to Akio Kawauchi for his helpful comments.
Citation
Tomomi Kawamura. "Integral region choice problems on link diagrams." Osaka J. Math. 60 (4) 835 - 872, October 2023.
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