11 problems

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

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

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 $M=o9_{41176}$ have the standard triangulation described in the source, with edges $C_1,\ldots,C_5$ and vertex surfaces $N_5,N_7,N_8,N_{10},N_{14},N_{16}$. The notation…

Let $M=o9_{41176}$. Write $\atil_M(n)=a_M(n+1)$ for its connected essential-surface count, and let $\phi(n)$ be Euler's totient function. Totient-plus-one conjecture. For $n\geq2$,…

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…

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

Let $L$ be a link in $S^3$ with $ll$ components, let $M$ be its exterior, and let $(\lambda_j,\mu_j)$ be the canonical longitude and meridian on the $j$-th boundary torus. For a pr…