An ¥-unknotting (unlinking) operation is defined by S. Jablan (1998) . Every self-crossing point of a KL can be turned into an uncrossing by introducing a "mirror". This can be repeated until we obtain the unknot (link without self-crossings). Analogously as we defined unknotting (unlinking) number, it is possible to define an ¥-unknotting (unlinking) number u¥ (K) by the "classical" and "standard" definition, to prove that two definitions are equivalent, and make a Conjecture on u¥ number: 

  1. u¥ (K) = 0, where K is the unknot (link without self-crossings); 
  2. u¥ (K) = min u¥ (K' )+1, where the minimum is taken over all minimal projections of knots (links) K', obtained from a minimal projection of K by an ¥-change. 

Every ¥-change transforms an alternating knot to an alternating knot, so the set of all alternating knots is closed with regard to ¥-changes. According to Tait's Flyping Theorem, all minimal projections of an alternating knot give the same result, so for every alternating knot it is sufficient to use only one minimal projection. 

The LinKnot function NoSelfCrossNo (webMathematica NoSelfCrossNo) calculates the u¥ number of a KL given by its Conway symbol, Dowker code, or P-data according to this Conjecture. 

Again, a particular value of the u¥ number is a characteristic of a family : for the knot family (2k+1), (k ³ 1), u¥ ((2k+1)) = 1; for the knot family  (2k) 2, (k ³ 1), u¥ ((2k) 2) = 2; for the knot family ((2k) (2l)), (k,l > 1), u¥ ((2k) (2l)) = 3, etc.

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