35 problems

Two-factor generation conjecture. For two-factor models, the ideal $I_{p,2}$ is generated by the $3\times 3$-minors and pentads.

Uniqueness conjecture. Every nonsingular homogeneous polynomial is nice. For degree greater than $2$, there is precisely one choice of coordinates in which a nice polynomial reache…

Let $L$ be an oriented link, let $n$ be a positive integer, let $GLZ^n_L(\tau,t)$ denote the invariant associated with the quantum superalgebra $U_q[\operatorname{osp}(2|2n)]$, and…

Let $L$ be an oriented link, let $m,n$ be positive integers, let $LG^{m,n}_L(\tau,t)$ denote the Links–Gould invariant, and let $\Delta_L$ denote the Alexander polynomial. Generali…

Let $K$ be a knot, let $\sg(K)$ denote its signature, let $D=\dt(K)$ be the determinant, let $H_1$ denote the first homology group of the double branched cover of $K$, and let…

The conjecture that ${\cal C}_3$ and ${\cal C}_4$ differ.

The conjecture that ${\cal C}_2$ and ${\cal C}_3$ differ.

Cubic Tutte relation conjecture. The second derivative of Speyer's polynomial satisfies

4-edge-twist conjecture. Then

3-edge-cut conjecture. The Speyer polynomials satisfy

Let $M_n$ be the algebra of all $n\times n$ matrices over a field $\mathbb{F}$, and let … be the algebra of invariants of an $m$-tuple under algebra automorphisms. Artin's conjectu…

Let $c(P)$ be the alternative measure of a polynomial, and let $k\geq 2$. Consider the integer-exponent family $1+x+x^{n_2}+x^{n_3}+\cdots+x^{n_k}$ and take the iterated limit with…

Alternative-measure conjecture.

Sign conjecture. The sign of $\lambda_{\mathcal{NCP}}(J_n)$ is

For each positive integer $d$, let ${\sf RSK}_{1^d,1^d}$ be the RSK operator indexed by the compositions $1^d$, and let ${\sf RSK}_{m,n,d}$ be the corresponding operator in dimensi…

Normalized flat Jones–Krushkal polynomial conjecture. For any almost classical flat knot $\alpha$, the normalized flat Jones–Krushkal polynomial satisfies

Optimality conjecture. For every almost encompassing polynomial $f$, one has

Let $K$ be a knotoid diagram. Let $K_1$ and $K_2$ be the starred knotoid diagrams obtained from $K$ by placing a star in the region incident to the tail and the region incident to…

Let $p$ be the prime occurring in the construction, let $n=\frac{p-3}{2}$, and let $a_k\in\mathbb{Z}[x,y]$ be the coefficients in the polynomial expression for $E_3$, with…

Let $d>1$ and let ${\bf k}$ be a field with characteristic greater than $d$ or characteristic zero. For a polynomial $Q\in {\bf k}[x_1,\ldots,x_s]$ of degree $d$, write…

Let $G$ be a finite group, let $p$ be a prime, let $P$ be a Sylow $p$-subgroup of $G$, and set $N=N_G(P)$. Let $D_G^{p'}(x)$ denote the $p'$-degree polynomial obtained from the deg…

The polynomiality criterion. The invariant ring $S(V)^G$ is a polynomial ring if and only if both of the following conditions hold: $S(V)^{G_U}$ is a polynomial ring for each subsp…

Completeness conjecture. One might conjecture that these are all such semi-invariant forms for $A$.

Oura's conjecture. The following analogous properties hold for $Th(\widetilde{\varphi_\ell^{G_{\rm{II}}}})$: (1) all zeros lie on a segment of the circle…

Let $P$ be a lattice polytope, and let $l^*(P;t)$ denote its local $h^*$-polynomial. Stanley's conjecture. The coefficients of $l^*(P;t)$ are non-negative. Karu proved this conject…