Kalfagianni–Tran's Strong Slope Conjecture

Let $K$ be a knot in $S^3$ with maximum colored-Jones degree

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

Set

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

For a Jones slope $p/q\in js(K)$ with $q>0$, say that it satisfies $SS(n)$ if there is an essential surface $F_n$ in the exterior of $K$ such that $F_n$ has boundary slope $4a(n)=p/q$ and

$$2b(n)=\frac{\chi(F_n)}{|\partial F_n|q}.$$

Such an essential surface is called a Jones surface.

Kalfagianni–Tran's Strong Slope Conjecture. For any knot in $S^3$, every Jones slope satisfies $SS(n)$ for some $n\in\mathbb{N}$.

This conjecture strengthens the Slope Conjecture by predicting the Euler characteristic of an essential surface from the linear term of the colored-Jones degree. It is established for various knot families but remains open for arbitrary knots.