1.9  Unlinking numbers 

The question of unknotting and unlinking numbers, or Gordian numbers is one of the most difficult in knot theory (Wendt, 1937; Nakanishi, 1981; Kawauchi, 1996; Kohn, 1991, 1993). In order to calculate unknotting and unlinking numbers, we need some link surgery. In every crossing of a KL it is possible to make a crossing change: to transform an overcrossing to undercrossing or vice versa

The unlinking number u(L) of a link L is the minimal number of crossing changes required to obtain an unlink from the link L; the minimum is taken over all projections of L. 

There are two different approaches for obtaining the unlinking number of L: 

  1. according to the classical definition, one is allowed to make an ambient isotopy after each crossing change and then continue the unlinking process with the newly obtained projection; 
  2. the standard definition requires all crossing changes to be done simultaneously in a fixed projection. 
Those two definitions are equivalent (see, e.g., C.Adams, 1994). 

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