The classification of rational subtangle replacements between rational tangles

Baker, Kenneth L; Buck, Dorothy

doi:10.2140/agt.2013.13.1413

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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

DOI: 10.2140/agt.2013.13.1413

Abstract

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”.

Keywords

rational tangle, tangle replacement, branched cover

Mathematical Subject Classification 2010

Primary: 57M27

References

Publication

Received: 3 May 2012

Revised: 31 October 2012

Accepted: 6 December 2012

Published: 30 April 2013

Authors

Kenneth L Baker
Department of Mathematics University of Miami PO Box 249085 Coral Gables, FL 33146 USA
http://math.miami.edu/~kenken
Dorothy Buck
Department of Mathematics Imperial College London South Kensington London SW7 2AZ UK
http://www2.imperial.ac.uk/~dbuck/

Volume 13, issue 3 (2013)