**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 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
[0]-type, [1]-type, or [¥]-type, respectively.
**

**If n is the number of crossings of an R-tangle, for every n
we obtain 2**^{n-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.