An n-move on a KL is a local change, where the remaining part of the KL remains unchanged. We say that two KLs, L1 and L2, are n-move equivalent if one can be transformed to the other by a finite number of n-moves and their inverses (-n-moves) (Przytycki, 2003). 

Nakanishi's Conjecture    Every knot is 4-move equivalent to an unknot (1979). 
 

The smallest known candidate for a counterexample to Nakanishi's Conjecture is the link 

2049953*.-1:.-1:.-1.-1:::-1::-1.-1
with n = 20 crossings. After all 3-moves applied to it and the obtained links are reduced, the number of crossings remains at least 20. 

Just as with the unknotting (unlinking) number u(L) and ¥-unknotting (unlinking) number u¥ (L), a minimal number of n moves required to obtain a knot or link k (k £ [n/2]) from a given KL can be defined as an n-move unknotting (unlinking) number un(L), taken over all projections of a KL. In fact, in the language of n-moves, unknotting number u(L) is u2(L), and u¥ (L) is u1(L). In the case of un(L) we can make a Conjecture analogous to Bernhard-Jablan Conjecture, and work only with minimum projections at all levels of unknotting (unlinking) process. According to such a Conjecture, the minimal number of n-moves necessary to obtain a knot or link k (k £ [n/2]) from a rational KL given by its Conway symbol, for a given parameter n, can be calculated by the LinKnot function NMoveRat (webMathematica NMoveRat). 

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