28 problems

Let $kge 2$, let $\beta\in \mathfrak{Br}_n$ be a braid, and let $L(\beta)$ denote its link closure. Write $\mathrm{H}_{0|k}(\beta)$ for the homology constructed in the paper and…

Let $G$ be a directed graph and let $E$ be a subset of its edge set $E(G)$. Write $\mathscr{H}_0^\mathbb{Q}(G)$ for the degree-zero Khovanov-Rozansky homology over $\mathbb{Q}$ and…

Let $\kappa$ be a link, let $n>0$, and let ${{\mathscr{H}}_{{(n)} {\scriptscriptstyle{\mathrm{symp}}}}^k}(\kappa)$ denote the link invariant constructed from Lagrangian Floer theor…

HOMFLY homology differentials conjecture. There is a homology theory $\mathcal H_*$ categorifying the HOMFLY polynomial, with differentials $\{d_N\}$ satisfying those three axioms,…

Let $K$ be a knot, and let $\overline{\mathit{KhR}}_N(q,t)$ denote the graded Poincaré polynomial of the $sl(N)$ Khovanov–Rozansky homology. A finite polynomial has only finitely m…

Let $C\subset\mathbb{C}^2$ be a singular curve with a planar singularity at the origin, and let $L$ be the algebraic link obtained by intersecting $C$ with a sufficiently small $3$…

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…

Let $m,n$ be coprime positive integers, let $\mathfrak{h}$ be the $(n-1)$-dimensional standard representation of $S_n$, and let $\mathrm{L}_{\frac{m}{n}}$ be the unique finite-dime…

Let $\bar{Y}_f$ be the enhanced $y$-ified Khovanov--Rozansky module of the link $L_f$. Let $\widetilde{Q}_{S}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-modul…

Let $L_f$ be the link of the plane curve germ $f=0$. Let $\widetilde{\operatorname{Quot}}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-module assembled from the…

Let $f(x,y)=0$ be a plane curve germ as above, let $\bar{X}_f(a,q,t)$ be the normalized Khovanov--Rozansky homology polynomial, and let $\Psi$ be the map from the relevant symmetri…

Let $\gamma$ be an unramified element with associated link $L_{\gamma}$, let $\mathrm{Sp}_{\gamma}$ be its affine Springer fiber, and let $\mathrm{gr}_P$ be the specified perverse…

Rank conjecture. The total rank of the symmetric $\mathfrak{gl}_1$-homology is equal to the total rank of the reduced triply graded homology.

Let $\beta$ be a braid on $n$ strands. Let $\operatorname{Hilb}^n(\mathbb{C}^2,\mathbb{C})$ be the indicated Hilbert scheme, let $\mathcal{T}$ be its tautological rank-$n$ bundle,…

Let ${\bm{b}}=(b_1,\ldots,b_m)$ and let $T(m,m;{\bm{b}})$ be the ${\bm{b}}$-colored $(m,m)$-torus link. Colored torus-link parity conjecture. The link $T(m,m;{\bm{b}})$ is parity;…

Let $a_1,\ldots,a_m,b\geq0$ and set $\bm{a}=(a_1,\ldots,a_m)$. A one-sided twisted complex of threaded digons is a chain of complexes obtained by increasing the number of crossings…

Let $\mathbf{K}$ be a framed, oriented knot, and let $\chi\mathbf{K}$ denote the knot with framing increased by one. For $b\geq0$, consider the directed system whose terms alternat…

Gorsky–Hogancamp–Simental–Rasmussen conjecture. The knots $\hat\beta(\mathbf{a}(f))$ and $\hat\beta_f$ are isotopic. Up to a monomial in $q$ and $t$,

Let $K$ be a knot and let $\overline{\mathrm{HHH}}(K)=\bigoplus \overline{\mathrm{HHH}}_{i,j,k}(K)$ be its reduced triply graded homology, where $i$ is the $a$-grading, $j$ is the…

Equality conjecture. Under these assumptions,

Gorsky–Neguț–Rasmussen conjecture. There is an isomorphism of bigraded vector spaces

Let $\operatorname{Tr}$ be the universal dg monoidal trace, let $\operatorname{Tr}(\mathbbm{1})$ be the trace of the monoidal unit, and let $X$ be a twisted complex built out of su…

Dowlin's graded spectral sequence conjecture. For each $n$, there are spectral sequences

Let $N$ and $M$ be positive integers, write $d=\frac{\gcd(N,M)}{}$, $N=dn$, and $M=dm$ with $n$ and $m$ coprime, and define … Let $P_{n,m}$ be the operator from the coprime case, a…

Let $u,v$ be binary words of compatible lengths, let $L_v(x;q,t)$ be the link symmetric function, and let $\widetilde{L}_v(x;q,t)=(1-q)^{n-|v|}L_v(x;q,t)$. Let $\widetilde{H}_\mu$…