System of tangle equations N(X+T_1)=L_1, N(X+T_2)=L_2 model DNA recombination and are related to the Nugatory Crossing and Cosmetic Surgery Conjectures. Our Uniqueness of Rational Solutions Conjecture states that a rational solution to a system of (framed) tangle equations is unique. We show that a failure of this purely topological statement would disprove the Jones Unknot Conjecture. Double branched covers turn the unframed version of the conjecture into one about lens space surgeries, leading to new results and conjectures about them, and potentially guiding towards counterexamples to the Jones Unknot Conjecture.