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2. Basic terms

In this paper will be considered only prime alternating knots and links, denoted by Conway notation [14]. For the both of them we will use the general term "link", considering knots as 1-component links. A prime link with singular digons, expressed by a Conway symbol, is called generating. Any other link could be derived from some generating link, by replacing singular digons by chains of digons. All links that could be derived from a generating link by such replacement make a family [19]. All links are distributed into disjoint sets, called by A.Caudron worlds [15].

Every 4-regular planar edge-connected graph uniquely defines an alternating link. If digonal edges are denoted by colored (bold) lines [37], then among bicolored graphs (including uncolored ones) representing generating links, we distinguish regular and combined graphs: 3-regular (where in each 3-valent vertex there is exactly one colored edge), 4-regular, and combined 3- and 4-valent graphs. A generating link is called basic if its bicolored graph is regular. If it is 4-regular, such graph is a basic polyhedron [12, 14]. The term "graph of link" means "bicolored graph of link". In the first part of this paper are considered only the graphs described by Conway notation (or canonical projections of links), and after that all reduced alternating projections.

The most powerful tool in our consideration is the symmetry [20]. It could be visualized representing links by graphs, and trying to imagine them as 3D ilustrations - maps on a sphere. For every alternating link we could distinguish the symmetry group G of its bicolored graph, and its subgroup G' of index 2 (or antisymmetry subgroup G/G'), obtained by alternating, representing the actual symmetry of a link.

 

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