16 problems

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

Minimum-degree Slope Conjecture. For any knot $K\subset S^3$,

Minimum-degree Strong Slope Conjecture. Every $p^*/q^*\in js^*(K)$ satisfies $SS^*(n)$ for some $n\in\mathbb{N}$.

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

Ichihara's conjecture. For all knots $K$,

Ishikawa--Shimokawa conjecture. Then

Torus-knot boundary-slope conjecture. Nontrivial torus knots are the only knots in $S^3$ with two boundary slopes.

Let $K$ be a hyperbolic knot in $S^3$ that admits non-trivial exceptional surgeries. A NIT boundary slope is a boundary slope that is non-integral or toroidal. For $b e{}$ an integ…

Let $K$ be a hyperbolic knot in $S^3$. A boundary slope is a slope represented by the boundary curves of an essential embedded surface in the exterior of $K$, and an exceptional su…

The Slopes Conjecture. The Jones slopes of a knot $K$ are boundary slopes.

Two slopes conjecture. If

Let $K\subset S^3$ be a knot, with Jones slopes $js_K$ and $js^*_K$ obtained as the cluster points of the normalized maximal and minimal degrees of its colored Jones polynomials. L…

HOMFLY slope conjecture. For any knot $K$,

Let $K\subset S^3$ be a knot. Let $J_K(n,t)$ be its colored Jones polynomial, and let $j(n)$ and $j'(n)$ denote its highest and lowest degrees in $t$. Define … A cluster point is a…

Let $K$ be a prime non-torus knot and let $E(K)$ be its exterior. A properly embedded orientable incompressible and boundary incompressible surface in $E(K)$ may have a boundary sl…

Let $K$ be a 2-bridge knot, and suppose that its boundary-slope set has exactly four distinct elements. A 2-bridge knot $K_1$ is minimal with respect to the Silver–Whitten partial…