119 problems

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…

Let $L$ be a prime alternating knot, and let $s(L)$ and ${\text{\it Kh}}'(L)$ be the integer and Laurent polynomial from Khovanov's structural conjecture. The signature of $L$ is t…

Let $b$ be a braid with a fixed number of strands, let $\widehat{b}$ denote its closure, and let the number of crossings of $b$ be the input size. Przytycki–Silvero conjecture. Com…

Let $\mathrm{HHH}_{S^n}(O_1)$ be the $S^n$-colored triply graded Khovanov–Rozansky homology of the unknot, with even generators $u_0,\ldots,u_{n-1}$ and odd generators…

Polynomial-growth conjecture. The total dimension satisfies

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

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…

Greedy matching conjecture. The greedy matching is a Morse matching for all torus braids, that is, the graphs

Let $P_n(t,q)$ denote the colored Khovanov Poincaré polynomial used in the paper, let $\mathcal{S}_0^2(\overline{\mathbb{CP}^2};h;\mathbb{Q})$ denote the skein lasagna module of…

Let $n\in\mathbb{N}_0$ denote the color of the 1-framed unknot, and let the formulas referred to as the formulas from Observation 1 hold for the colored Khovanov homology in color…

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…

Let $K$ be a thin non-alternating knot, and let $\sigma(K)$ and $s(K)$ denote its signature and Rasmussen invariant, respectively. Signature–Rasmussen conjecture. Their absolute va…