An algorithm that changes the companion graphs (Q1842621)

The projection \(P(K)\) of a knot (link) \(K\) is a diagram obtained by mapping the knot (link) under orthogonal projection of \(\mathbb{R}^3\) onto \(\mathbb{R}^2\), where all intersections are transversal. For a given projection, shade its regions white and black so that the unbounded region is white. Let \(G^b_{P(K)}\) \((G^w_{P(K)})\) denote the pseudograph with the vertices corresponding to the black (white) regions of \(P(K)\) and the edges corresponding to the crossings shared by the black (white) regions of \(P(K)\). The graphs \(G^b_{P(K)}\) and \(G^w_{P (K)}\) are called companion graphs. The author introduces three types of graph moves corresponding to Reidemeister moves for diagrams and proposes an algorithm that, for an arbitrary alternating knot (link) \(K\), changes the graph \(G^b_{P(K)}\) (or \(G^w_{P(K)})\) to its companion graph by the use of that graph moves.