## 1.2.1  Gauss and Dowker code

The 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 self-intersecting 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

 {{1,2,2,3,4},{2,1,1,4,3},{3,1,2,4,4},{4,3,3,2,1},

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 figure-eight knot (knot 41 in the classical notation, where a symbol of the form nij 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 Alexander-Briggs notation) (Alexander and Briggs, 1926-27; 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

 {{1,2,6,5,3},{2,3,4,6,1},{3,1,5,4,2},{4,3,5,6,2},{5,3,1,6,4},{6,2,4,5,1}},
beginning from 1 and (1,2), then from 1 and (1,6), and finally from 2 and (2,6) we obtain the Gauss code
 {{1,2,4,5},{1,6,4,3},{2,6,5,3}},
so it is a shadow of the well known 3-component link- Borromean rings 623