### 1.9.1  Unlinking gap

Work on those problems can get started by experimenting with rational KLs given in Conway notation, using the LinKnot functions AllStates Rational and UnR. Certainly, those problems can be extended to all KLs, and not restricted only to rational KLs. In this case, instead of the very fast functions AllStates Rational and UnR that use for a computation continued fractions and work only with rational KLs, you can use the general (but, unfortunately, much slower) LinKnot functions fAllStatesProj and UnKnotLink.

The functions AllStatesRational and fAllStatesProj give as the first output datum the unknotting (unlinking) number uM(L) of a fixed KL projection M. The function UnKnotLink gives as the result an estimated unlinking number uE. If for a knot K holds uE(K) = [(s(K))/ 2], or for a link L holds uE(L) = [(s(L)+1)/2], where s is the signature, the result is the exact unknotting (unlinking) number u(L). From the numbers uM(L) and u(L) we calculate the projection unlinking gap dM(L) = uM(L)-uE(L).