The LinKnot functions RK (webMathematica RK), R (webMathematica R), and RL (webMathematica RL) calculate the number of rational knots, rational KLs, and rational links for a given number of crossings n, respectively.

The LinKnot function RatSourceKLNo (webMathematica RatSourceKLNo) calculates the number of rational source KLs with n crossings according to the general recursive formula: 

b[0] = 1,   b[1] = 1,   b[2n-2]+b[2n-1] = b[2n], 
b[2n]+b[2n-1]-f[n-1] = b[2n+1], 
where f is the Fibonacci sequence given by the recursion 
f[0] = 1,   f[1] = 1,   f[n-2]+f[n-1] = f[n].

For n ³ 4 we obtain the sequence 1, 1, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410 ..., known as the sequence A001224. Both of these sequences, A005418 and A001224, have been discovered before, but in a different context, related to "Binary grids" and "Packing a box with n dominoes". The actual results are rational source KLs given in the following table: 


 
 
n = 2 
No. of KLs: 1 

 
 
n = 4 
2 2 
No. of KLs: 1 

 
 
n = 5 
2 1 2 
No. of KLs: 1 

 
 
n = 6 
2 2 2  2 1 1 2 
No. of KLs: 2 

 
 
n = 7 
2 2 1 2  2 1 1 1 2 
No. of KLs: 2 

 
 
n = 8 
2 2 2 2  2 1 2 1 2  2 2 1 1 2  2 1 1 1 1 2 
No. of KLs: 4 

 
 
n = 9 
2 2 1 2 2  2 2 2 1 2  2 1 2 1 1 2  2 2 1 1 1 2  2 1 1 1 1 1 2 
No. of KLs: 5 

 
 
n = 10 
2 2 2 2 2  2 1 2 2 1 2  2 2 1 1 2 2  2 2 1 2 1 2  2 2 2 1 1 2
2 1 1 2 1 1 2 2 1 2 1 1 1 2  2 2 1 1 1 1 2 2 1 1 1 1 1 1 2
No. of KLs: 9 

 
 

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