120 problems

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Surjectivity conjecture. The homomorphism $\Sigma$ is surjective.

Gorsky–Oblomkov–Rasmussen conjecture. The stable Khovanov homology $\operatorname{Kh}(T(n,\infty))$ is the dual to the homology of the Koszul complex associated to the sequence…

Przytycki–Sazdanović conjecture. The Khovanov homology of a closed $3$-braid can only have torsion of order $2$.

Let $L\subset S^3$ be a link. The singly graded symplectic Khovanov cohomology is denoted by $Kh^k_{symp}(L)$, and the bigraded combinatorial Khovanov homology by $Kh^{i,j}(L)$. Se…

Let $K$ be a knot whose integral Khovanov homology $Kh(K,\mathbb{Z})$ is supported on diagonals … where $w$ is the width of its Khovanov homology, and let $\sigma(K)$ denote its si…

Mirror-duality conjecture. The diagram

Let $K$ be a knot. Its Khovanov homology over $\mathbb{Q}$ is a bigraded vector space, and write $s$ for Rasmussen's invariant. A pawn move piece is … and a knight move piece is ……

Rasmussen's spectral sequence conjecture. There is a spectral sequence from $\overline{Kh}(K)$ to $\widehat{\mathit{HFK}}(K)$.

Let $L$ be a link, and let $c$ and $c'$ be components of $L$. Denote by $\widetilde{E}^k(L,c)$ the reduced Szabó spectral sequence associated to $L$ with distinguished component…

Let $U$ denote the unknot, and let $K$ be a prime knot. Suppose that ${\widehat H}_{red}(K)$ and ${\widehat H}_{red}(U)$ are isomorphic as $\delta$-graded vector spaces over…

Let $S$ be a compact oriented surface and let $J,\widetilde J$ be representations of $S$ by sequences of link diagrams, with endpoint diagrams $J_0,J_1$ and…

For each natural number $m$, let $T_{2m}$ be the full projective tilting module in the relevant parabolic category ${\mathcal O}$ for $\mathfrak{sl}_{2m}$, and let ${\mathcal H}_m$…

Let $D_{p,p}$ and $D_{p,p+1}$ be the torus-knot diagrams used in the paper, and let $H^{i,j}(D)$ denote their Khovanov homology in homological degree $i$ and quantum degree $j$. No…

Owens' H-thin conjecture. For any H-thin knot $K$,

Let $G$ be a graph, and let $H^{**}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with the algebra ${\mathcal A}_m$. A graph is loopless if it has no loops, and a cyc…

Let $L$ be a transverse representative of an alternating smooth link. Write $\psi(L)\in\widetilde{Kh}(L)$ for the transverse-link invariant in reduced Khovanov homology, let…

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…

Rank–torsion conjecture. The knots $K_1$ and $K_2$ have the same ranks of the Khovanov homology if and only if they have the same torsion.

Reduced-torsion conjecture. The knot $K$ is T-rich if and only if its reduced Khovanov homology has torsion.

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

Powers-of-two torsion conjecture. The Khovanov homology of every link has no torsion of order $p$ for any $p$ other than a power of $2$.

Torsion existence conjecture. Except for the trivial knot, the Hopf link, and their connected sums, the Khovanov homology of $L$ has $2$-torsion.

Let $\kappa$ be an oriented link, represented as the closure of an $m$-stranded braid, and let $\mathit{Kh}_{\mathrm{symp}}^*(\kappa)$ denote the graded symplectic invariant define…

Let $K$ be a knot, and let $\mathcal{C}(K)$ be its complex of graded modules over the algebra $\mathcal{A}$. The knot is H-restricted when the non-acyclic indecomposable summands o…