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6. *Unknotting number*
More complicated is the question of
unknotting numbers [10, 31-35]. For
the calculation of unknotting numbers we accepted
the folowing **Conjecture**:
(a) *u*(1) = 0, where 1 is the unknot;
(b) *u*(*k*) = min *u*(*k*^{-})+1,
where the minimum is taken over all knots *k*^{-}, obtained
from a minimal projection of *k* by one change of crossing.
Because by one change of crossing in a knot *pq*
we could obtain only knot (*p*-2)*q *or *p*(*q*-2),
we propose that for the unknoting numbers of knots belonging to the family
*pq* holds the recursion formula: *u*(*pq*) = min (*u*((*p*-2)*q*),
*u*(*p*(*q*-2)))+1. |