In this way, we can easily determine type of any R-tangle. For example, for the R-tangle 3 5 4 1 3 we have the 0-1 code 11011, and its type is []:


1 11 = [1] [1] 110 = [0] [0] 1101 = [] [1] 11011 = [1] [1]
[1] [0] [] [1] [0]


In order to simplify notation, in the case of R-tangles the brackets [ ] will be omitted, and the products of the tangles [0] and [1] will be expressed as 0-1 sequences. E.g., 011 means [0] [1] [1], etc.

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 R-tangle not beginning or ending with 2, it can be expressed by 0-1 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 2-component link (obtained from [0]). Hence, every rational KL is a knot, or 2-component link.

Stellar (pretzel) tangles are obtained from R-tangles by using the operation of ramification: (a,b)=-a-b=a 0+b 0 and consist of at least three R-tangles. 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 R-tangles 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 R-tangles 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,,] 2-component links, and for [,,] 3-component links (Fig. 1.75). For pretzel KLs with four R-tangles, knots are obtained for [0,1,1,1], [1,1,1,], [0,1,1,], [0,0,0,1], [0,0,1,], [0,0,0,], 2-component links for [1,1,1,1], [0,0,1,1], [1,1,,], [0,1,,], [0,0,0,0], [0,0,,], 3-component links for [1,,,], [0,,,], and 4-component links for [,,,]. In general, for pretzel KLs with t R-tangles there are ((t+2) || 2) type sets. Among them, t+2 will give knots, 2t-2 will give 2-component links, and the number of i-component link types will be t-i+1 (i=3,4,,t).

 

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