On region crossing change and incidence matrix

Abstract

In a recent work of Ayaka Shimizu [5] , she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams.In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals to c − n + 1, here c denotes the size of the graph.

An Integral Region Choice Problem on Knot Projection

“…Here we define an integral matrix A i (D), called a region choice matrix, as a coefficient matrix of the system of equations of region choice problem. Cheng and Gao proposed an incidence matrix in [2], this is a modulo 2 reduction of A 1 (D). In Section 3, 4, we show the main result of this paper.…”

When is region crossing change an unknotting operation?

“…In this section we will take a quick review of incidence matrix which was defined in [2]. Given a link diagram D, we can colour all the regions of D with white and black such that each pair of regions which share some common boundaries have different colours, i.e.…”

“…If we work with Z 2 coefficients, then it is not difficult to find that the incidence matrix M(D) is closely related to region crossing changes. THEOREM 2·1 ( [2]). Moreover, given a set of regions, in order to understand the effect of region crossing changes on these regions, one just needs to read the positions of 1's in the sum (with Z 2 coefficients) of the corresponding row vectors.…”

“…In [2], we gave a complete answer to this question for 2-component links. In general, region crossing change is not always an unknotting operation for link diagrams.…”

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.…”