28 problems

Let $T_{3,m}$ be a torus knot, and let $\mathcal P(T_{3,m})$ denote its reduced superpolynomial. For $m=3k+1$ or $m=3k+2$, define the three polynomials…

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…

Links-Quivers Correspondence for homology. The colored HOMFLY-PT homologies of $L$ may be represented by chain complexes such that

Let $L$ be an arbitrary link, let $\mathrm{HY}(L)$ denote its homology, and let $\mathcal{E}_k$ and $\mathcal{F}_k$ be the operators acting on it. Let $\mathcal{H}_2$ be the Lie al…

Algebraic generation conjecture. The homology of $d_W$ is generated by $u_i$ and $\mu_k^W$ as an algebra. The supplied text establishes that the coefficients $\mu_i^W$ are cycles,…

Let $u_i$ and $\xi_j$ be the generators in the polynomial algebra underlying the differential $d_N$, and for $k=1,\ldots,n-1$ define … The differential satisfies…

Let $L$ be a Hopf link in $\mathbb{R}P^3$ whose components are colored by an $n$-dimensional skew-symmetric and/or $m$-dimensional symmetric representation $R$ of $su(2)$. Speciali…

A sequence of chain complexes of filtered graded vector spaces is said to be homologically $q$-holonomic when it satisfies the proposed notion of homological $q$-holonomicity. For…

Let $K\subset S^3$ be a knot, let $\mathcal{A}$ be either $\mathbb{F}_p$ or $\mathbb{Q}$, and let $\mathcal{C}$ denote the knot concordance group. Concordance-invariance conjecture…

For $G=GL_n$, let $gammabin g derbrace{llbracket trrbracket}{\mathcal O}$ be regular semisimple, let $cbeta=\beta(\gamma)$ be the associated braid, and let…

Let $K$ be a knot admitting the standard knots-quivers correspondence, so that the node variables satisfy $x_i\sim x$. Let $Q$ be a quiver corresponding to $K$, with one spectator…

Generalized-quiver partition-function conjecture. After this substitution, the generalized-quiver partition function equals the generating function of Gromov-Witten invariants coun…

Connectivity conjecture. Any two balanced pairings of the same size $n$, where $n$ is odd, are connected.

Flattening conjecture. The flattening of ${\mathcal{V}}$ is a complex ${\mathcal{E}}$ satisfying the requirements of Gorsky–Negut–Rasmussen's conjecture.

Cube-compatible identification conjecture.

Dunfield–Gukov–Rasmussen conjecture. There is a spectral sequence from HOMFLY-PT homology to knot Floer homology.

Manolescu's conjecture. There is an isomorphism

Rectangular DAHA-superpolynomial positivity conjecture. The coefficients of $H\!D_{r,s}(j\omega_i;q,t,a)$ are positive integers. After the displayed transformation, the resulting p…

Let $K$ be a knot, and let $H^{\Lambda^r}_{i,j,*}(K)$ and $H^{S^r}_{i,j,*}(K)$ denote, respectively, its antisymmetric- and symmetric-coloured HOMFLY homology groups, with indices…

The generation conjecture. The homology of $d_N$ is generated as an algebra by $x_k$ and $mu_k$. This gives a proposed algebraic description of stable $\operatorname{sl}_N$-homolog…

Functoriality conjecture. This assignment of a map to a cobordism is independent of the choice of Morse function, i.e. this makes the knot homology theory $\mathcal{K}(-)$ functori…

Let $Y({\underline{i}})$ be the affine-Grassmannian variety associated with ${\underline{i}}=({\0},1^k,{\u2})$ for some $k\in\mathbb N$, and let … be the projection forgetting the…

Let $\mathcal{A}_{n,m}$ be algebras represented on the stable homology space $\mathcal{H}(T_{n,m})$, and let $e_1,\ldots,e_n$ denote the even polynomial generators and…

Let $\mathcal{P}_{s}^{2k}(T_{n,m})(q,t)$ denote the relevant coefficient of the triply graded Poincaré polynomial, and let the sum range over all marked Dyck paths in the…

Dunfield–Gukov–Rasmussen conjecture. There exists a homology theory with the stated Euler-characteristic, differential, and symmetry properties such that for all $N>1$ the homology…