1.2.1 Gauss and Dowker codeThe possibility of studying knots from the mathematical point of view was first proposed by C.F.Gauss. Gauss made codes of immersed curves by assigning letters to the crossing points of a selfintersecting curve and determined "words" defining a closed curve. This problem is completely solved by M.Dehn (Dehn, 1936)Let a labelled plane graph L' be given. By applying to it the Component Algorithm, and just reading the labels of vertices visited, we obtain the Gauss code of L': the list of vertex sequences divided into circuits. For example, from the graph given by the plane embedding adjacency list
beginning from the vertex 1 and the second oriented edge (1,2), we obtain the Gauss code {{1,2,4,3,2,1,3,4}}. All edges of the graph are exhausted in one circuit, so our graph is a shadow of a knot, namely the shadow of a figureeight knot (knot 4_{1} in the classical notation, where a symbol of the form n_{i}^{j} denotes ith KL with n crossings and j components, and for knots the upper index j = 1 is omitted; for knots, this notation is known as the AlexanderBriggs notation) (Alexander and Briggs, 192627; Rolfsen, 1976). Beginning from from the same vertex by the first oriented edge (1,2) we obtain the Gauss code {{1,2,3,4,2,1,4,3}}, and so on. In a knot shadow, every choice of new beginning point and oriented edge incident to it results in a new Gauss code. From the octahedron graph
