|A third operation: ramification
is defined as (a,b) = -a-b.
Some results obtained by applying the operations mentioned above are illustrated in the corresponding figures.
Connecting in pairs NE and NW, and SE and SW ends of a tangle we obtain a numerator closure; connecting in pairs NE and SE, and NW and SW ends we obtain a denominator closure. A rational tangle is any tangle obtained by the operation of product on elementary tangles. A rational KL is a numerator closure of a rational tangle.
The rational knots are also known as 2-bridge knots, Viergeflechte, or 4-plats. Their were first considered from the mathematical point of view by O.Simony (1882, 1884), and classified by H.Schubert (1956). J.Conway proposed the beautiful idea that rational KLs are related to continued fractions (Conway, 1970; Kauffman and Lambropoulou, 2002).
A tangle is called algebraic if it can be obtained from elementary tangles by the operations of product and sum. KL is algebraic if it is a numerator closure of an algebraic tangle.
A 4-valent graph without bigons is called a basic polyhedron. For every KL shadow, its basic polyhedron can be identified by collapsing bigons till none of them remains. In fact, most of the basic polyhedra are real geometrical polyhedra- 3-vertex connected 4-regular graphs, but in the list of basic polyhedra are also included some 2-vertex connected graphs. More precisely, the basic polyhedra are 4-regular 4-edge-connected, at least 2-vertex connected plane graphs. The basic polyhedron 1* is illustrated in the corresponding figure. KLs that have at least one shadow which reduces by bigon collapse to 1* are called 1*-links or algebraic KLs. All others are called polyhedral KLs. The three operations with tangles mentioned are sufficient for the notation of algebraic KLs. Polyhedral KLs need special notation. A KL obtained from a basic polyhedron P* by substituting tangles t1, ..., tk in appropriate places is denoted by P*t1...tk, where the number of dots between two successive tangles shows the number of omitted substituents of value 1. For example, 6*2:2:2 0 means 6*22.214.171.124.2 0.1, and 6*2 1.2.3 2:-2 2 0 means 6*2 1.2.3 2.1.-2 2 0.1.