When is region crossing change an unknotting operation?

Abstract

In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also discuss the relation between region crossing change and the Arf invariant of proper links.Comment: 10 pages, 7 figure

Region crossing change on surfaces

“…Let K be a knot diagram, and R a region of R 2 \ K (for simplicity, we also call R a region of K), then applying region crossing change on R yields a new knot diagram, which is obtained from K by reversing all the crossing points incident to R. Figure 2 indicates how to convert the trefoil knot into the unknot by applying region crossing change on region R. In fact, not only this diagram of trefoil knot can be transformed into a diagram representing the unknot. Let K be a knot diagram and S a set of some crossing points in K. Following [5], we say S is region crossing change admissible if one can obtain a new knot diagram K ′ from K by a finite sequence of region crossing changes, here K ′ is obtained from K by switching all the crossing points in S. As the main result of [13], Ayaka Shimizu actually proved the following result. It follows immediately that region crossing change is an unknotting operation for knot diagrams.…”

“…A natural question is when region crossing change is an unknotting operation for link diagrams. This question was answered by the second author in [5].…”

The multi-region index of a knot

“…In this standard diagram, each of the four regions have both crossings in their boundary, and thus changing any region flips the diagram to its mirror image. Cheng and Gao [CG12] and Cheng [Che13] classified the link diagrams for which region crossing change is an unlinking diagram. Dasbach and Russell [DR18] examined region crossing changes for link diagrams on oriented surfaces.…”

Region crossing change on spatial-graph diagrams

“…1, a region crossing change on the region R changes the crossings c 1 , c 2 and c 3 , and a region crossing change on S changes d and e. It was shown in [5] that any crossing change on a knot diagram can be realized by a finite number of region crossing changes. For a link diagram, it was proved in [3] (see also [2]) that any crossing change at a self-crossing of a knot-component, and any pair of crossing changes at non-self-crossings for any two knot-components, can be realized by region crossing changes.…”