In this way, we can easily determine type of any Rtangle. For example, for the Rtangle 3 5 4 1 3 we have the 01 code 11011, and its type is [¥]:
For k=1 there are two sequences: 0 of the type [0] and 1 of the type [1]; for k=2 four sequences: 00 and 01 of the type [¥], 10 of the type [1], and 11 of the type [0]; for k=3 eight sequences: 000, 010, 101 of the type [0], 001, 100, 011 of the type [1], and 010, 101 of the type [¥]; for k=4 sixteen sequences: 0000, 0001, 0100, 0101, 1010, 1011 of the type [¥], 0010, 1000, 0110, 1101, 1111 of the type [1], and 0011, 1001, 1001, 1100, 0111, 1110 of the type [0], etc. Because every rational KL is obtained as a numerator closure of an Rtangle not beginning or ending with 2, it can be expressed by 01 sequence. This means that it contains only tangles of the type [0] or [1], so as the final result can be obtained only a tangle of the type [1], [¥], or [0]. It can be only a knot (obtained as a numerator closure from [1], [¥]), or 2component link (obtained from [0]). Hence, every rational KL is a knot, or 2component link. Stellar (pretzel) tangles are obtained from Rtangles by using the operation of ramification: (a,b)=ab=a 0+b 0 and consist of at least three Rtangles. Pretzel KLs are numerator closures of pretzel tangles. For denoting the types of pretzel knots double brackets will be also omitted in the following sense: the type of pretzel knot obtained from Rtangles of the type [0], [1], [¥], will be shortly denoted as [¥,1,0], since [0]0=[¥], [1]0=[1], [¥]0=[0]. For example, the type of the pretzel tangle 2,3,2 1 will be [[0],[1],[¥]] = [¥,1,0]. For pretzel KLs with three Rtangles we have 10 possible type sets. For [1,1,1], [1,1,¥], [0,0,1], [0,1,¥], [0,0,¥] are obtained knots, for [0,1,1], [1,¥,¥], [0,0,0], [0,¥,¥] 2component links, and for [¥,¥,¥] 3component links (Fig. 1.75). For pretzel KLs with four Rtangles, knots are obtained for [0,1,1,1], [1,1,1,¥], [0,1,1,¥], [0,0,0,1], [0,0,1,¥], [0,0,0,¥], 2component links for [1,1,1,1], [0,0,1,1], [1,1,¥,¥], [0,1,¥,¥], [0,0,0,0], [0,0,¥,¥], 3component links for [1,¥,¥,¥], [0,¥,¥,¥], and 4component links for [¥,¥,¥,¥]. In general, for pretzel KLs with t Rtangles there are ((t+2)  2) type sets. Among them, t+2 will give knots, 2t2 will give 2component links, and the number of icomponent link types will be ti+1 (i=3,4,¼,t).
