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A natural generalization of a crossing change is a rational subtanglereplacement (RSR). We characterize the fundamental situation of the rationaltangles obtained from a given rational tangle via RSR, building on work ofBerge and Gabai, and determine the sites where these RSR may occur. In additionwe also determine the sites for RSR distance at least two between 2-bridgelinks. These proofs depend on the geometry of the branched double cover.Furthermore, we classify all knots in lens spaces whose exteriors aregeneralized Seifert fibered spaces and their lens space surgeries, extendingwork of Darcy-Sumners. This work is in part motivated by the common biologicalsituation of proteins cutting, rearranging and resealing DNA segments --effectively performing RSR on DNA `tangles'.

The Classification of Rational Subtangle Replacements between Rational Tangles