On region crossing change and incidence matrix

TL;DR

This paper investigates the conditions under which region crossing change unknots links and introduces an incidence matrix for link diagrams, establishing a relation between the matrix's rank and the diagram's components.

Contribution

It characterizes when region crossing change unknots 2-component links based on linking number parity and relates the incidence matrix rank to the number of link components.

Findings

Region crossing change unknots 2-component links if and only if linking number is even.

The incidence matrix's rank determines the number of components in a link diagram.

A signed planar graph represents an n-component link if the incidence matrix rank equals c - n + 1.

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.