241 problems

Let $K$ be a knot. The unreduced Khovanov homology $Kh(K;\mathbb{Z})$ is the Khovanov homology of $K$ with integer coefficients, and $2$-torsion means an element of order $2$. Shum…

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, let $s(K)$ denote Rasmussen's invariant from Lee homology, and let $\tau(K)$ denote Ozsváth–Szabó's invariant from Heegaard Floer homology. Rasmussen's conjectur…

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

Let Khovanov homology be the homological invariant of links categorifying the Jones polynomial, and let link cobordisms be smooth cobordisms between links. Khovanov's functoriality…

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…

Knight Move Conjecture. The Khovanov homology of the closure of every $3$-strand braid over $\mathbb{Q}$ is the direct sum of one pawn move piece and several knight move pieces.

Rasmussen's rank conjecture. The rank of reduced Khovanov homology is always greater than or equal to the rank of knot Floer homology:

Mirror-duality conjecture. The diagram

Diagram-independence conjecture. The spectral sequence in Theorem 1.3 should be independent of the planar diagram $\mathcal{D}$.

Greedy matching conjecture. For any tangle diagram $T$, the greedy matching $M_{\operatorname{gr}}$ is a Morse matching on the delooped Khovanov complex…

Thin surgery discrepancy question. Does there exist a strongly invertible L-space knot $(K,\Phi)$ with a thin surgery such that $\varkappa(K,\Phi)$ is not thin?

Let $D_L$ be a link diagram, and let $KH^{i,j_{\rm min}}(D_L)$ denote its extreme Khovanov homology in the minimal quantum grading. Przytycki–Silvero conjecture. The extreme Khovan…

No-gap conjecture. If $L$ is a prime quasi-alternating link that is not a $(2,n)$-torus link, then the differential grading of $\mathcal{H}(L)$ has no gap.

Fix an even integer $n$, and consider the weaving knot $W(n+1,m)$, where $m$ is coprime to $n+1$. Along either of the two lines specified in Theorem 3, consider the normalized Khov…

Let $L_1$ and $L_2$ be links, let $L_1\amalg L_2$ be their disjoint union, and let $\mathcal{X}_\mathit{Kh}(L)$ denote the Khovanov homotopy type of a link $L$. Disjoint-union conj…

Let $L$ be a link, let $m(L)$ denote its mirror, and let $\mathcal{X}_\mathit{Kh}^j(L)$ denote its Khovanov homotopy type in quantum grading $j$. Mirror duality conjecture. The spa…

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…

Let $D$ be a plane diagram of a link, let $C(D)$ be the associated complex of graded $\mathbb Z[c]$-modules, and let quasi-isomorphisms between such complexes be associated to Reid…

Let $D$ be a generic plane projection of an oriented link $L$ in $\mathbb R^3$, and let ${\cal H}^{i,j}(D)$ be the finitely generated bigraded groups constructed from $D$, whose is…

A virtual link is a link represented in the virtual-link setting, and a twisted Khovanov complex is a version of the Khovanov complex adapted to twisted virtual links. Khovanov–Ras…

Let $L$ be a link and let $KKH(L)$ denote its Khovanov homology, with torsion meaning the torsion subgroup of these abelian groups. The 2-torsion conjecture. It was conjectured tha…

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 ${\mathcal H}_m$…