If in the standard definition we take only minimal projections instead of working with all projections, we cannot always obtain the correct unlinking number. This is illustrated by the well known example of the knot 108 (or 5 1 4 in Conway notation), given by Y. Nakanishi (1983) and S. Bleiler (1984). The rational knot 5 1 4 has only one minimal projection. According to the standard definition, unknotting takes at least three crossing changes (in the crossings denoted by circles). If we apply the classical definition: make a crossing change in the middle point of the diagram followed by the reduction 5 -1 4 = 3 1 2, we obtain the minimal projection of the knot 3 1 2 that can be unknotted by only one crossing change. As a result we obtain the correct unknotting number 2. The same unknotting number will be obtained if we take the non-minimal projection of the knot 5 1 4 and use the standard definition. As was shown by J. Bernhard (1994), the same property holds for the complete knot family (2k+1) 1 (2k), k ³ 2. 

Using the classical definition, we can obtain the correct unknotting number u(5 1 4) = 2 from the minimal projection of the knot 5 1  4. Therefore, we propose the following conjecture: 
 

Bernhard-Jablan Conjecture

  1. u(L) = 0 for any unlink L; 
  2. u(L) = min u(L-)+1, where the minimum is taken over all minimal projections of links L-, obtained from a minimal projection of L by one crossing change.
This means that we take a minimal projection of a KL, make a crossing change in every vertex, and minimize all the KLs obtained. The same algorithm is applied to the first, second, ... k-th generation of the KLs obtained. The unlinking number is the number of steps k in this recursive unlinking process. 


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