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

Abstract

We study systems of two-tangle equations $$ \begin{align*}\begin{cases} N(X+T_1)=L_1,\\ N(X+T_2)=L_2, \end{cases}\end{align*} $$ 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 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.