28 problems

Chi-ribbon closure conjecture. Then $L_1$ is an alternating link. In particular, $K_1$ is an alternating knot in the knot case.

Ribbon cobordism converse. If $\Lambda(L_2)$ admits a cubiquitous embedding into a stabilization of $\Lambda(L_1)$, then there exists a ribbon cobordism from $\Sigma(L_1)$ to…

Let $L$ be a link with $c$ components. Call $L$ Khovanov-basic if its Khovanov complex $\operatorname{Kh}(L)$ is homotopy equivalent to a direct sum of basic complexes and exactly…

H-thin implies T-thin conjecture. Every H-thin link is T-thin. In particular, every non-split alternating link is T-thin.

Let $L$) be the planar Murasugi sum of special alternating links $L_1,\ldots,L_n$. Murasugi sum conjecture. Then … This conjecture was proposed in connection with the analogous pro…

Defect proportion comparison conjecture. For all even $c\geq 4$ and all odd $c\geq 11$, the proportion of $c$-crossing prime alternating links $L$ with $\gamma(L)>\Gamma(L)$ is lar…

Crosscap-three classification conjecture. For each even crossing number $2k\geq 6$, there are exactly

Exponential decay conjecture. Both proportions exponentially decay with respect to crossing number:

2d log-concavity conjecture. The absolute values $(|a_{ij}|)$ form a 2d log-concave sequence with no interior zeros. This conjecture is proposed as a bidimensional analogue of log-…

Strong version of Ishii's conjecture. The coefficients satisfy $a_{ij}=(-1)^{\mu+i+j+1}|a_{ij}|$ for all $i,j$, and $(a_{ij})$ has no interior zeros. The source notes that this str…

Let $K$ be an alternating hyperbolic link in $S^3$. Its hyperbolic volume is denoted by $\operatorname{vol}(K)=\operatorname{vol}(S^3\setminus K)$, and its determinant by $\det(K)$…

Greene–Owens cubiquity conjecture. $\Sigma(L)$ bounds a rational homology $4$-ball if and only if $\Lambda(L)$ is cubiquitous.

Let $D$ be a reduced alternating diagram of a knot $K$ with determinant $p$, where $p$ is prime. A Fox $p$-coloring is a coloring of the arcs of $D$ by elements of $\mathbb Z_p$ sa…

Let $\mathcal{L}$ be a $\mathbb{Z}^2$-periodic alternating link, with toroidally alternating quotient link $L$. Let $P(z,w)$ be the characteristic polynomial of the toroidal dimer…

Greene's conjecture. If a pair of links have homeomorphic branched double-covers, then either both are alternating or both are non-alternating.

Two-minima one-saddle conjecture. If $L$ bounds a ribbon surface with two minima and a single saddle point, then $L$ is algorithmically ribbon.

Let $\mathcal L$ be a hyperbolic biperiodic alternating link, and let $\{X(L_n)\}$ be the associated sequence of finite-volume covers described above. Write $TH_1(X(L_n))$ for the…

Let $K$ be a hyperbolic alternating link, and let $B$ be an augmented unknot around any two parallel strands of $K$. Augmentation volume conjecture. … This conjecture would give an…

Let $\mathcal{L}$ be any biperiodic alternating link, with toroidally alternating $\Lambda$-quotient link $L$. Let $p(z,w)$ be the characteristic polynomial of the toroidal dimer m…

Let $\mathcal L$ be a biperiodic alternating link with alternating quotient link $L$, and let $K_n$ be alternating hyperbolic links satisfying…

Let $\mathcal L$ be a biperiodic alternating link with alternating quotient link $L$, and let $K_n$ be alternating hyperbolic links satisfying…

A banding of the unknot is a band surgery on the unknot that produces an alternating link. An almost alternating diagram of the unknot is an alternating diagram with one dealternat…

Let $K$ be a knot in $S^3$. An alternating surgery on $K$ is a non-trivial Dehn surgery whose result is the double branched cover of a non-split alternating link. McCoy's alternati…

Let $K$ be an alternating hyperbolic link, let $\operatorname{vol}(K)$ denote its hyperbolic volume, and let $\det(K)$ denote its determinant. Alternating volume–determinant conjec…

Let $K$ be an alternating hyperbolic knot, and let $K'$ be obtained from $K$ by changing crossings in a diagram. The alternating-diagram maximal volume conjecture. (a) If any cross…