1.11.1 Tangle types

Regardless of the sign (+1, -1), there are two elementary tangles,  and , where for  we can distinguish between two different positions  and [¥] L. Kauffman and S. Lambropoulou (2002a).

A sequence of positive integers not beginning by 1 will be called R-tangle, where the space between numbers denotes the tangle product. If an R-tangle consists of k numbers (k ³ 1), we say that its length is k. All algebraic KLs will be obtained from R-tangles by applying two tangle operations- product and ramification.

For R-tangles of an odd length the first bigon (or the chain of bigons) is always taken as horizontal, and for R-tangles of an even length as vertical. Every R tangle consists of two strands connecting SW-SE and NW-NE ends, SW-NE and SE-NW ends, or SW-NW and SE-NE ends. According to this, every tangle will be of the -type, -type, or [¥]-type, respectively.

If n is the number of crossings of an R-tangle, for every n we obtain 2n-2 R-tangles. For example, for n=2 we have R-tangle 2 of the type ; for n=3 two R-tangles: 3 of the type , and 2 1 of the type [¥]; for n=4 four R-tangles: 4 and 3 1 of the type , 2 2 of the type [¥], and 2 1 1 of the type . By taking every number modulo 2, we obtain 0-1 sequences of the length k. As numerator closure of tangles of the types  and [¥] we obtain knots and from  2-component links.

Knowing that the product of tangles a and b is defined as a b = -a+b = a 0+b, where a 0 denotes the tangle a reflected in SE-NW mirror line, we deduce simple product rules for tangle types:

   = [¥]   = [¥]  [¥] = [¥,¥] =[¥2]   =    =   [¥] = [¥] [¥]  =  [¥]  =  [¥] [¥] = [¥]

The right multiplications by  form a dihedral symmetry group with the invariant point , and the right multiplications by  the cyclic group of order 3.

The LinKnot function fTangleType (webMathematica fTangleType) calculates the type of R-tangle, giving as the result 0,1,2 for a tangle of the type ,, or [¥], respectively. The function fMakeType (webMathematica fMakeType) produces all the tangles of given type with n crossings.