Every rational unlink
can be expressed by one of the following Conway symbols
c_{0}
c_{1} ... c_{r1} c_{r} (1) 1 (c_{r}1)
c_{r1 }... c_{1} c_{0} 

c_{0}
c_{1} ... c_{r1} (c_{r}1) 1 (1) c_{r}
c_{r1} ... c_{1} c_{0}, 

where c_{i} ³
0 for i = 0,...,r and c_{r} 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
The number of such links
is 0 for every even n, and for odd n it is given by the formula
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.
