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

We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio ${T}_q\in \mathbb Q(q)$ofany$2$-tangle$T$andweconjecturethatfor$q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets $\{T{\}}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.