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The classification of rational subtangle replacements between rational tangles

Kenneth L Baker and Dorothy Buck

Algebraic & Geometric Topology 13 (2013) 1413–1463

A natural generalization of a crossing change is a rational subtangle replacement (RSR). We characterize the fundamental situation of the rational tangles obtained from a given rational tangle via RSR, building on work of Berge and Gabai, and determine the sites where these RSR may occur. In addition we also determine the sites for RSR distance at least two between 2–bridge links. These proofs depend on the geometry of the branched double cover. Furthermore, we classify all knots in lens spaces whose exteriors are generalized Seifert fibered spaces and their lens space surgeries, extending work of Darcy and Sumners. This work is in part motivated by the common biological situation of proteins cutting, rearranging and resealing DNA segments, effectively performing RSR on DNA “tangles”.

rational tangle, tangle replacement, branched cover
Mathematical Subject Classification 2010
Primary: 57M27
Received: 3 May 2012
Revised: 31 October 2012
Accepted: 6 December 2012
Published: 30 April 2013
Kenneth L Baker
Department of Mathematics
University of Miami
PO Box 249085
Coral Gables, FL 33146
Dorothy Buck
Department of Mathematics
Imperial College London
South Kensington
London SW7 2AZ