21 problems

Let $\beta$ be a braid whose closure $L(\beta)$ has one connected component, written $\pi_0(L(\beta))=1$. Let $\mathrm{HH}_{m|m}(\beta)$ denote the corresponding homology, graded b…

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…

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 $\Gamma$ 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…

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…

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 $L$ 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…

Specialization conjecture. Under the algebraic specialization $\wp(x)=x^n$, the groups

$H_{1-1}$–$\mathit{HFK}_{2}$ conjecture. The total homology $H_{1-1}(K)$ is isomorphic to $\mathit{HFK}_{2}(K)$.

Let $B^{(m)}$ be the $m$-colored version of a braid diagram $B$, and let $H_N(B^{(m)})$ denote its colored $athfrak{sl}(N)$ link homology. Denote by $H$ the colored HOMFLY-PT link…

Let $L$ be a non-split link and let $k$ be a fixed positive integer. Let $n_k(L)$ be the number of distinct $k$-crossing link diagrams for $L$ up to 2-space isotopy. Let $N$ be the…

Let $k$ be the number of crosscuts, let $N$ be the number of copies of Khovanov homology in the diagramless homology of a link with $k$ crosscuts, and let the minimum crossing numb…

For a crossing with colours $i$ and $j$, let $d_k^{\pm}$ be the successive differentials in the associated complex, with $0\leq k\leq i-2$. Differential-square-zero conjecture. … f…

Let $N\geq 4$ and consider the $\mathfrak{sl}(N)$-link homology defined in the preceding sections over the rational theory, with its relations and homomorphisms suggested to admit…