51 problems

Certain colored link homologies are associated with links and admit graded dimensions as numerical invariants. Gorsky–Oblomkov–Rasmussen conjecture. Their graded dimensions should…

Let $\Gamma$ be a planar trivalent graph, let $\langle\Gamma\rangle_0$ be the state space obtained from the specialized foam evaluation over $\mathbb{F}_2$, and let…

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…

For any link, let $\mathcal{F}(L)$ be the foam-based bigraded complex and let $\operatorname{KR}(L)$ be the KhovanovRozansky complex. Suppose the already established isomorphism be…

Consider any other evaluation of closed foams satisfying the same relations as the evaluation used in the thesis, and let it produce a functorial link homology categorifying the…

Let $k>1$ and let $L$ be an oriented $k$-component link. Let $\widetilde{\mathrm{H}}(L)$ denote the proposed redefined triply graded link homology theory. Finite-dimensional extens…

Let $D$ be a diagram of an oriented framed link $L$ whose components or strands are colored by numbers from $1$ to $n-1$, and let $P_n(L)$ be the corresponding colored link invaria…

Let $\Gamma$ be a labeled trivalent graph, let $H(\Gamma)$ be the cohomology of its associated matrix-factorization complex, and give the coefficient rings $S(e)$ the grading deter…

Let $\Gamma$ be a labeled trivalent graph, let $C(\Gamma)$ be the associated tensor-product $2$-complex of matrix factorizations, and let $H(\Gamma)$ be its cohomology. When loops…

Let $H_3$ denote the doubly graded homology theory constructed from the matrix-factorization complexes in the paper, and let the homology theory constructed in [Kh3] be tensored wi…

Categorification conjecture. For each simply-laced $\mathfrak{g}$, the invariant $\bracket{L}$ admits such a categorification, with $\bracket{L}$ as its Euler characteristic:

Let $\mathcal{L} = \mathcal{L}_1 \sqcup \dots \sqcup \mathcal{L}_d$ be a link with $\mathcal{L}_1$ an unknot and suppose $p \in \mathcal{L}_1$. For colors $k_1,\dots,k_d$, let…

Chain-level $\mathfrak{gl}_N$ functoriality conjecture. The chain-level $\mathfrak{gl}_N$ link invariant arises as the truncation of an $\mathbb{E}_3$-monoidal functor

Let $0\leq d_1\leq\cdots\leq d_1$, let $\beta=\operatorname{JM}_n^{d_1}\ldots\operatorname{JM}_2^{d_{n-1}}$, and set $J=J(d_1,\ldots,d_n)$, where … The notation…

For integers $0\leq d_1\leq d_2\leq\cdots\leq d_n$, define the Coxeter braid … where $\operatorname{JM}_k$ is a Jucys–Murphy element on the first $k$ strands. A braid is parity whe…

Let equation denote the paper's preceding conjectural relation, and let $n$ and $k$ be the rank and generator index appearing there. Higher-rank trace conjecture. Equation also hol…

Let $\beta\in\mathrm{Br}_m$ and let equation denote the paper's stated trace/decategorification equality between type $B$ spin link homology and the spin-colored…

Let $\beta\in\mathrm{Br}_m$ be a braid, and let the type $B$ spin link homology and the spin-colored $\mathfrak{so}_{2n+1}$ link polynomial be the invariant and polynomial defined…

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 $C$ be a convex monotone curve, let $L_C$ be its associated link, and let $\mathcal P^{\mathcal E}_C(a,q,t)$ and $\mathcal P^{\operatorname{KR}}_{L_C}(a,q,t)$ denote respective…

Let $C$ be a monotone curve, let $L_C$ be its associated link, and let the construction of the paper assign invariants to $C$. A link is algebraic when it arises as an algebraic li…

A monotone link is the projection to ${\mathbb R}^2/{\mathbb Z}^2$ of a curve $C$ from $(0,0)$ to $(m,n)$ whose two coordinates are monotone increasing. A curve is convex if it is…

Let $L$ be an $n$-component link in a homology 3-sphere, and let $\boldsymbol{\mathbf{n}}$ be a choice of framing. Let $\mathit{CLI}^-(L,\boldsymbol{\mathbf{n}})$ denote the associ…

Let $G$ be a split reductive group with Weyl group $W$, root lattice $\Phi$, and $V=\mathbf{Z}\Phi\otimes_{\mathbf{Z}}\mathbf{Q}$. Let $\mathit{Br}_W^+$ denote the positive braid m…

Let $G$ be a complex, connected, reductive algebraic group, let $B_+$ and $B_-$ be opposed Borel subgroups, and let $U_+$ and $U_-$ be their unipotent radicals. Let…