Rational link fertility

Abstract

Abstract A knot K is the resultant of a knot H if there exists a minimal crossing diagram D of H such that some crossings of D can be altered to produce K . H is fertile if every prime knot K with crossing number less than c ( H ) is a resultant of H . H has fertility number n if every prime knot K with crossing number less than or equal to n is a resultant of H . We classify the fertility and fertility number of all rational links. This requires the introduction of the analogous concept of link fertility. We find $$2^2_1$$ 2 1 2 , $$4^2_1$$ 4 1 2 , $$5^2_1$$ 5 1 2 , $$6^2_2$$ 6 2 2 , $$6^2_3$$ 6 3 2 , and $$7^2_2$$ 7 2 2 are fertile and $$6^2_1$$ 6 1 2 is not. We establish that almost all two-component links have fertility number greater than or equal to five.