According to Tait's
First Conjecture (Murasugi 1996, Theorem 11.5.5), proved independently in 1986
by L.Kauffman, K.Murasugi and M.Thistlethwaite (Kauffman, 1987b; Murasugi, 1987a,b;
D. Garity extended the one-parameter Bernhard family (2k+1) 1 (2k) (k ³ 2) of knots to the two-parameter family (2k+1) (2l+1) (2k) (k ³ 2, l ³ 0, k > l) with the unknotting gap d=1, and discovered the first two-parameter family of rational links (2k) 1 (2l) (k ³ 2, l ³ 2) with an arbitrarily large unlinking gap (Garity, 2002). He proved that for the knot family (2k+1) (2l+1) (2k) the result obtained from a fixed minimal projection according to the standard definition is k+l+1, and the exact unknotting number is k+l, so the unknotting gap is always 1. For the links belonging to the family (2k) 1 (2l), the unlinking number obtained from a fixed minimal projection is k+l-1, and the exact unlinking number is £ l, so the unlinking gap is at least k-1 and can be made arbitrarily large for a sufficiently large k.
A distance of links L1 and L2 is a minimal number of crossing changes in L1 required to obtain L2, the minimum taken over all projections of L1. A distance of link projection L'1 from a link L2 is a minimal number of crossing changes in the particular fixed link projection L'1 required to obtain a projection of L2. The LinKnot function AllStatesRational (webMathematica AllStatesRational) calculates all states of a given projection of a rational KL given by its Conway symbol, that is, all possible choices of signs in a given projection of a rational KL. The first datum in the result is the unknotting (unlinking) number obtained from the fixed projection of a given rational KL (according to the standard definition). It is followed by the list of minimum numbers of crossing changes necessary to obtain certain rational KL given by its Conway symbol, i.e., the list of rational KLs obtained from a given projection by all possible crossing changes. The function AllStatesRational is very fast, because it works with continued fractions. A similar, but much slower function fAllStatesProj (webMathematica fAllStatesProj), based on the function ReductionKnotLink works with all KLs, giving as an output the unknotting (unlinking) number obtained from the fixed projection of a given KL, followed by a list of minimal distances of KLs obtained by crossing changes and their reduced P-data.
By using the functions
you can easily check the Nakanishi-Bleiler example
(the knot 5 1 4), or similar examples from the family (2k+1) 1 (2k). In
all those cases, the results obtained according to the standard definition,
by using a fixed minimal projection, will give an incorrect unknotting
number k, and the function UnR will give the correct unknotting number
k-1. This example can also be extended to links belonging to the family
(2k) 1 (2k) (k > 1) and their unlinking numbers. A fixed minimal projection
of a link (2k) 1 (2k) can be unlinked with minimum 2k-1 crossing changes,
and the correct unlinking number obtained according to the Conjecture,
calculated by the function UnR, is k. In our example the difference between
those two numbers, the unlinking gap, is k-1 and can be made arbitrarily
large for sufficiently large k.