A period of a knot projection is the order of rotation that transforms it to itself. The period of the knot 2 2 is 2, and the same holds for all knots of the family (2p) (2q) (p ³ q ³ 2). For the invertibility of achiral knots we have one more criterion: an achiral knot is invertible if its period is 2; otherwise, it is non-invertible. 

The LinKnot function RationalAmphiK (webMathematica RationalAmphiK) calculates the number and Conway symbols of all rational achiral knots for a given number of crossings n. Calculating the number of achiral knots for n = 2k (k = 1, 2, 3,...) we obtain the Jacobsthal sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, ...  defined by the recursion 

a[1] = 0,   a[2] = 1,  2a[k]+a[k+1] = a[k+2]
and given by the general formula 
a[n] =  2n - (-1)n
3
.

All rational links are 2-component. Since all oriented alternating links with an even number of components are chiral (Cerf, 1997), there are no oriented rational achiral links. For 2 £ n £ 12, achiral non-oriented rational links are: 


 
 
 
n = 2 
n = 6 
3 3 
n = 8 
2 1 1 1 1 2 
n = 10 
5 5  3 2 2 3  2 2 1 1 2 2 
n = 12 
4 1 1 1 1 4  3 1 1 1 1 1 1 3  2 3 1 1 3 2 2 1 3 3 1 2 
2 1 1 2 2 1 1 2 
n = 14 
7 7  5 2 2 5  4 2 1 1 2 4  3 4 4 3
3 2 2 2 2 3  3 1 2 1 1 2 1 3 2 4 1 1 4 2 2 2 3 3 2 2
2 2 1 2 2 1 2 2  2 1 2 1 1 1 1 2 1 2  2 1 1 1 1 1 1 1 1 1 1 2 

 

Their achirality can be tested by using the computer program SnapPea by J. Weeks. 

A rational non-oriented link is achiral iff its Conway symbol is symmetrical and has an even number of crossings (Kauffman and Lambropoulou, ). The LinKnot function RationalAmphiL (webMathematica RationalAmphiL) calculates the number and Conway symbols of all rational achiral non-oriented links for a given number of crossings n. After Hopf link 2 (212) for n = 2, calculating the number of achiral non-oriented rational links for n = 2k (k = 2, 3, 4, ...) we obtain again the Jacobsthal sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, ...

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