83 problems

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

Manolescu's conjecture. There is an isomorphism

Colored Kauffman homology conjecture. There exist quadruply graded colored Kauffman homologies with rich structural properties.

Let $T_{m,n}$ be a torus knot, and let $\bar{\operatorname{rCS}}_{\lambda}(T_{m,n};a,q,t)$ denote its stable large $N$ refined Chern–Simons invariant, where the color $\lambda$ is…

Let $\mathcal{H}^{\mathrm{HOMFLY}}(K)$ denote HOMFLY homology of a knot $K$, and let $\mathcal{H}^{\mathfrak{sl}(N)}(K)$ denote its $\mathfrak{sl}(N)$ homology. Let $d_N$ be the co…

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 $K$ be a knot in $S^3$. Let $\mathcal{K}$ be an $n$-crossing braid diagram for $K$ with its outermost strand cut, and let $G_I$ for $I\in\{0,1\}^n$ be the planar trivalent grap…

Let $L$ be an alternating link, and let $\mathcal{H}(L)$ be its bigraded homology theory. The Jones polynomial and the signature of $L$ are the data under consideration. Alternatin…

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 two-bridge knot. Call $K$ $\mathcal H$-thin if all generators of its triply graded homology $\mathcal H(K)$ have the same $\delta$-grading, and let $S(K)$ denote the a…

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 $A$ be the complex of bracket star-diagrams, with its differential bialgebra structure, and let $H_*(\mathcal K)$ denote the homology bialgebra of the space $\mathcal K$ of lon…

Let $d$ be the ambient dimension, let $A$ be the complex of bracket star-diagrams equipped with the differential bialgebra structure constructed in the paper, and let…

For each knot $K$, consider a triply graded homology theory with graded pieces $\mathcal H^{\rm Kauffman}_{i,j,k}(K)$, and let $\{d_N\}$ be a family of differentials together with…

Let $K$ be a knot, and let $\mathcal F(K)$ denote a finite reduced polynomial in $\mathbb{Z}[\lambda^{\pm 1},q^{\pm 1},t^{\pm 1}]$. Let $\operatorname{HSO}_N(K;q,t)$ and…

Let $K$ be a knot. A finite polynomial $\overline{\mathcal F}(K)$ in $\mathbb{Z}[\lambda^{\pm 1},q^{\pm 1},t^{\pm 1}]$ is sought, with specializations to the $so(N)$ and $sp(N)$ kn…

Let $K$ be a knot and let $\mathcal{H}^{\bullet}_N\{+1\}-\mathcal{H}^{\bullet}_N\{-1\}$ denote the doubly graded complex obtained from the $sl(N)$ homology by shifting the $q$-grad…

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…

Let $\lambda$ be a partition, and let $\lambda^t$ denote its transpose. Consider the reduced colored HOMFLY homologies of a knot colored by $\lambda$ and by $\lambda^t$. GGS mirror…

Consider the decomposition of reduced colored HOMFLY homology into the homological blocks appearing in the GNSSS color-shifting framework. GNSSS homological-block differential conj…

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…

The $y$-ified projector $\mathbf{P}_n^y$ is a complex associated with $n$ strands satisfying the desired projector properties, including idempotence, sliding over and under crossin…

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…