Garoufalidis's Slope Conjecture for Jones slopes

Let $K$ be a knot in the $3$-sphere $S^3$. Its colored Jones function has maximum degree described, for sufficiently large $n$, by a quadratic quasi-polynomial

$$\delta_K(n)=a(n)n^2+b(n)n+c(n),$$

where $a(n),b(n),c(n)$ are rational-valued periodic functions, and define the set of Jones slopes by

$$js(K)=\{4a(n)\mid n\in\mathbb{N}\}.$$

A properly embedded surface in the knot exterior $E(K)=S^3-\operatorname{int}N(K)$ is essential if each component is orientable, incompressible, and boundary-incompressible. A rational number or infinity is a boundary slope if it is represented by the boundary of such a surface, and let $bs(K)$ denote the set of boundary slopes.

Garoufalidis's Slope Conjecture. For any knot $K$ in $S^3$, every Jones slope is a boundary slope:

$$js(K)\subset bs(K).$$

The conjecture relates the quadratic term of the colored Jones polynomial to the topology of the knot exterior. It is the quadratic part of the Strong Slope Conjecture and remains open in general, although it is known for many classes of knots.