The Classification of Rational Subtangle Replacements between Rational Tangles

TL;DR

This paper characterizes rational subtangle replacements (RSR) in rational tangles, identifies their possible sites, and classifies related knots and surgeries, with applications to DNA recombination processes.

Contribution

It provides a comprehensive classification of RSR in rational tangles and their implications for knot theory and DNA topology, extending previous work on tangles and lens spaces.

Findings

Characterized RSR sites in rational tangles.

Determined RSR distance at least two for 2-bridge links.

Classified knots in lens spaces with specific properties.

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-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'.