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We study systems of two-tangle equations which play an important role in the analysis of enzyme actions on DNA strands.
$$ \begin{align*}\begin{cases} N(X+T_1)=L_1,\\ N(X+T_2)=L_2, \end{cases}\end{align*} $$
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 two-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 two-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 and Ovsienko.
Additionally, we relate systems of tangle equations to the cosmetic surgery conjecture and the nugatory crossing conjecture.
