Non-hyperbolic solutions to tangle equations involving composite links

TL;DR

This paper solves specific tangle equations involving 2-bridge links and rational tangles, focusing on cases where the associated double branched cover is not hyperbolic or when the tangle is algebraic, with implications for DNA enzyme action models.

Contribution

It provides solutions to tangle equations under non-hyperbolic and algebraic tangle assumptions, extending previous work in the field.

Findings

Solved tangle equations with non-hyperbolic double branched covers.

Addressed tangle equations involving only 2-bridge links with algebraic tangles.

Identified conditions under which solutions exist for these tangle systems.

Abstract

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2 \# b_3$, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$ nontrivial. We solve this system of equations under the assumption $\widetilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e.$O$ is not $\pi$-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle.