Non-hyperbolic solutions to tangle equations involving composite links

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 [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are rational tangles, and [Formula: see text] is a 2-bridge link, for [Formula: see text], with [Formula: see text] and [Formula: see text] nontrivial. We solve this system of equations under the assumption [Formula: see text], the double branched cover of [Formula: see text], is not hyperbolic, i.e. [Formula: see text] is not [Formula: see text]-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption [Formula: see text] is an algebraic tangle.

Tangle equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals

“…The link L 2 is observable in the experiment. Since tangles T 1 and T 2 are known, DNA recombination processes lead to systems (1.1), where X is the unknown (see, e.g., [1,5,8,9,11,38,39,43,44] for further discussion of such systems).…”