For pretzel KLs consisting of more then three R-tangles, from the same type we can obtain several different orders of the particular symbols 0, 1, . For example, for the type [0,1,1,] there are two possible orders: [0,1,1,] and [0,1,,1], etc.

In general, the following rules hold: [1,1]=[0], [1,]=[], [a,0]=[a] for every a (a {0,1,}). The type set [,,] where occurs k times will be shortly written as [k] (k=1,2,). Moreover, the calculation of the reduced types of pretzel KLs is commutative on the symbols 0, 1, . Using these rules, every type set can be reduced to [0], [1], or [k].

The number of components of a pretzel KL of the reduced type [0] is 2, [1] are knots, and [k] k-component links (k=1,2,).
For example, [1,1,1,1,1,0,0,0,,] = [1,,] = [2], so it is a 2-component link. The symbol [k] represents a tangle of the type [] with k-1 already closed components. Hence, [k] = [k-1], where the subscript k-1 denotes the number of already closed components. A numerator closure of [k] = [k-1] is a k-component link.

In order to keep track of the number of closed components we add it, as a subscript, to the existing notation for tangle types. For example, [0k,l] = [k+l], [k] [l] = [k+l], [k] [0 l] = [0k+l], etc.. Knowing that the numerator closure of [1] and [] gives one component, and the numerator closure of [0] gives two components, we conclude that the numerator closure of [1k-1] and [k-1] is a k-component link, and the numerator closure of [0k-1] is a (k+1)-component link. Therefore, tangle type calculation gives the number of components of any algebraic KL.

The equality of tangle types is the equivalence relation, dividing the set of all R-tangles into three equivalence classes [0], [1], [] with the minimal representatives 2, 3, 2 1. Instead of the tangle 1, the tangle 3 is taken as the minimal representative of the type [1], to avoid ambiguity coming from two different orientations ("vertical" or "horizontal") of tangle 1.
We construct families of KLs adding an even number of bigons to already existing bigons or chains of bigons. Since this addition preserves tangle type, the number of components will be preserved inside families. In general, the number of components of a KL remains preserved if we replace any tangle of a given type by a tangle of the same type.

 

PreviousContentsNext