Every rational unlink can be expressed by one of the following Conway symbols 
c0 c1 ... cr-1 cr (-1) 1 (cr-1) cr-1 ... c1 c0
c0 c1 ... cr-1 (cr-1) 1 (-1) cr cr-1 ... c1 c0,
where ci ³ 0 for i = 0,...,r and cr ³ 2. 

The LinKnot function RatKnotGenU0 (webMathematica RatKnotGenU0) gives the number and the list of Conway symbols of all rational unknots with n crossings, and the function RatLinkU0 (webMathematica RatLinkU0) gives the same result for rational unlinks. The number of such knots is given by the formula 

2[[(n-2)/ 2]]-2.
The number of such links is 0 for every even n, and for odd n it is given by the formula 
2[[(n-7)/ 2]].

Next figure represents a knot with the unknotting number 1. You can try to find the appropriate vertex for the crossing change that results in unknot. If you don't believe that such a point exists, you can check its unknotting number by putting its Conway symbol 

3 2 1 1 2 3 3 1 1 2 3 2 1 1 2
in the LinKnot function UnR. In a similar way for the unlink 
3 2 1 1 2 3 3 -1 1 2 3 2 1 1 2 3
the answer will be 0, showing that this is an unlink. 

Another way to show this is to import KLs in question in KnotPlot, using the LinKnot function fCreateGraphics, and relax them. 

 
 

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