  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 $\{T\}_q\in \mathbb
Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ 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.
