54 problems

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 $\beta$ 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,…

Cautis and Kamnitzer construct a link homology using the derived category of coherent sheaves on certain flag-like varieties. Cautis–Kamnitzer homology conjecture. Their homology i…

Let $K\subset S^3$ be a knot, and let $\mathcal{H}(K)$ be the triply graded homology theory associated with the HOMFLY polynomial. For each positive integer $N$, the source conside…

Let $K\subset S^3$ be a knot, let $\overline{H}_N(K)$ denote its $sl(N)$ KhovanovRozansky homology, and let $P_K(a,q)$ be its normalized HOMFLY polynomial. The triply graded HOMFLY…

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…

Galashin--Lam's conjecture. Determine the curves $C$ for which

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 $\mathcal H^{daha}$ denote the DAHA superpolynomial and let $KhR$ denote the reduced colored Khovanov–Rozansky polynomial of an iterated torus link. DAHA–KhR coincidence conjec…

Let $Z=\{f(x,y)=0\}\subset\mathbb{C}^2$ be a curve with a unibranch singularity at $0$, and let $K=K_Z$ denote its link, which is a knot. Let $\delta$ be the Milnor number of…

Let $m,n$ be coprime natural numbers, set $c=\frac{m}{n}$, let $\mathfrak{h}$ be the $(n-1)$-dimensional standard representation of $S_n$, and let $\mathrm{L}_c$ be the unique fini…

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 $L_f$ be the link of the plane curve germ $f=0$, with $b$ components, and let $\mathbb{C}[\vec{x},\vec{y}]$ act on its $y$-ified Khovanov--Rozansky homology. Let…

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…

Let $gamma in \mathfrak{gl}_n((t))$ have reduced spectral curve, let $\mathrm{Sp}_{\gamma}$ be its affine Springer fiber, and let $L_{\gamma}$ be the link of the spectral curve sin…

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 $\operatorname{FHilb_n^{dg}}(\mathbb{C})$ be the dg flag Hilbert scheme parametrizing full flags of ideals containing an ideal of codimension $n$ in $\mathbb{C}[x,y]$, with suc…

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…

Let ${\bm{b}}=(b_1,\ldots,b_m)$ and $N=\sum_{i=1}^m b_i$. Let $T(m,m;{\bm{b}})$ be the ${\bm{b}}$-colored $(m,m)$-torus link, and let $E_{\bm{b}}$ be the algebra defined from the c…