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).
In order to make a more complete and detailed research of the unlinking gap, we wrote some additional LinKnot functions. LinKnot function UnRFixProj (webMathematica UnRFixProj) calculates unlinking number of a fixed projection for a given rational KL. The additional function fGapRat (webMathematica fGapRat) calculates the estimated unlinking number of a given rational link L, the unlinking number of its fixed minimal projection, detects rational KLs with an unlinking gap, and computes its value d(L). The calculation of unlinking gaps for rational KLs is very fast because the functions UnR and fGapRat work with continued fractions. Similar, but much slower functions UnKnotLink and fGap (webMathematica fGap), based on the function Reduction KnotLink, are used for the calculation of (projection) unlinking gaps for non-rational KL projections.
If the Conjecture is not true in general, the list of KLs with an unlinking gap given here will remain correct, because for every KL from that list the unlinking number obtained from a minimal projection uM(L) is greater than the estimated unlinking number uE(L) which represents the upper bound for the exact unlinking number u(L), i.e., uM(L) > uE(L) ³ u(L).
However, in that case our list may not be complete: KLs whose exact unlinking number is smaller than the estimated one will also have an unlinking gap.