11 problems

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. Consider $N_\Phi$ a Seifert fibered space with base orbifold $S^2(2,2,2,3)$ and gluing matrix $\Phi$ $=$ ……

Strong Cabling Conjecture. For any even integer $b\ge 6$, the triple $(0,b,1)$ is not realizable.

Let $F_K(x,q)$ be the $F_K$ invariant, with coefficient expansion $F_K(x,q)=x^{1/2}\sum_{n\geq 0}f_n(q)x^n$. For sufficiently large $n$, write the minimal $q$-degree as a quadratic…

Four-torus bound conjecture. There are at most four pairwise disjoint, non-isotopic, nested, embedded twice-punctured tori with an integral slope in the complement of $K$.

Let $M$ be a manifold satisfying the hypotheses of Theorem, and define the smoothed count … where $\atil_M(n)=a_M(n+1)$. Asymptotic conjecture. Either $\atil_M(n)=0$ for all suffic…

Let $B$ be the exterior of the knot $K13n586$, and let $\atil_B(n)=a_B(n+1)$ count connected essential surfaces of genus $n+1$. Let $\phi(n)$ denote Euler's totient function. Genus…

Strong Slope Conjecture. Given a Jones slope $\frac{p}{q}\in js_K$, with $q>0$ and $\operatorname{gcd}(p,q)=1$, there is an essential surface $S\subset S^3\setminus K$ with…

Strong Neuwirth conjecture. There exists an essential non-orientable spanning surface for $K$.

Neuwirth conjecture. There exists a closed surface $F$ such that $F$ contains $K$ non-separatingly and is essential in the exterior of $K$.

Lopez conjecture. There exists a small knot in every small closed $3$-manifold.

Planar surface conjecture. A non-meridional essential planar surface in a knot exterior is either a disk or an annulus.