1.11.1 Tangle types

Regardless of the sign (+1, -1), there are two elementary tangles, [1] and [0], where for [0] we can distinguish between two different positions [0] 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 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 [0]-type, [1]-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 [0]; for n=3 two R-tangles: 3 of the type [1], and 2 1 of the type []; for n=4 four R-tangles: 4 and 3 1 of the type [0], 2 2 of the type [], and 2 1 1 of the type [1]. By taking every number modulo 2, we obtain 0-1 sequences of the length k. As numerator closure of tangles of the types [1] and [] we obtain knots and from [0] 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:



[0] [0] = [] [0] [1] = [] [0] [] = [,] =[2]
[1] [0] = [1] [1] [1] = [0] [1] [] = []
[] [0] = [0] [] [1] = [1] [] [] = []


The right multiplications by [0] form a dihedral symmetry group with the invariant point [1], and the right multiplications by [1] 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 [0],[1], or [], respectively. The function fMakeType (webMathematica fMakeType) produces all the tangles of given type with n crossings.

 

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