19 problems

Let $\Delta$ and $\Delta'$ be two $h$-transverse polygons. A lattice-preserving transformation is an element of the affine group of $\mathbb{R}^2$ preserving the lattice…

Let $i\in\mathbb{Z}_{\geq 2}$ be the number of interior lattice points of a half-integral polygon, and require the polygon to have at least $2$ boundary lattice points. An Ehrhart…

Let $i$ be the number of collinear interior lattice points of a half-integral polygon, with $i\geq 3$. A polynomial enumeration conjecture. There are exactly … half-integral polygo…

Let $a(n)$ be the maximum of the sum of the squares of the edge-lengths over full grid lattice polygons with $n$ vertices, and let $\alpha(n)$ be the explicit lower-bound function…

Let $\mathbb T=\mathbb T_{M_1,M_2}$ be a lattice tube, and let $L$ be a non-split link embeddable in $\mathbb T$. Let $p_{\mathbb T,n}(L)$ denote the number of $n$-edge embeddings…

For a knot type $K$, let $p_n(K)$ be the number of $n$-edge polygons of knot type $K$. Let $0_1$ be the unknot, let $f_K$ be the number of prime knot factors in the knot decomposit…

Let $K$ be a knot type, and let $p_n(K)$ denote the number of $n$-edge lattice polygons of knot type $K$. Let $0_1$ denote the unknot, and let $f_K$ be the number of prime knot fac…

Mutation-equivalence conjecture. All consistent dimer models associated with the same lattice polygon are mutation equivalent.

Free-energy lower-bound asymptotic conjecture. The free energy is asymptotic to the lower bound as $f\to-\infty$ for every lattice tube $\mathbb T$. The corresponding asymptotic st…

Let $\mathbb T$ be a simple-cubic lattice tube and let $p_{\mathbb T,n}(K)$ denote the number of $n$-edge polygons in $\mathbb T$ with knot type $K$. Let $p_{\mathbb T,n}(0_1)$ den…

Let $\Delta\subseteq\mathbb{R}^2$ be a lattice polygon whose interior polygon $\Delta^{(1)}$ is two-dimensional and contains $g\geq 4$ lattice points. Assume that…

Let $g\geq 5$ satisfy $g\equiv 1\bmod 4$, and let $\Delta_g^n$ and $\Delta_g^m$ denote the lattice polygons indexed by integers $n$ and $m$ with $0\leq n,m\leq (g+3)/4$. A curve is…

Let $k$ be an algebraically closed field, let $f\in k[x^{\pm1},y^{\pm1}]$ be an irreducible Laurent polynomial with two-dimensional Newton polygon $\Delta(f)$, and let $C_f$ be the…

Let $k$ be an algebraically closed field, let $f\in k[x^{\pm1},y^{\pm1}]$ be an irreducible Laurent polynomial with two-dimensional Newton polygon $\Delta(f)$, and let…

Let $C/k$ be a curve, and let … where $\simeq$ denotes birational equivalence. Product-order minimum conjecture. The set $S_{1,1}(C)$ admits a minimum with respect to the product o…

Fix $m\in\mathbb{Z}_{>0}$. Consider toric surfaces $S$ with ample anticanonical divisor $-K_S$, without imposing conditions on their singularities, such that the anticanonical poly…

Let $\Delta\subset\mathbb{R}^2$ be a two-dimensional lattice polygon. For a Laurent polynomial $f\in\mathbb{C}[x^{\pm 1},y^{\pm 1}]$, write $\Delta(f)$ for its Newton polygon, and…

Let ${\varPi}_n$ be the ensemble of convex lattice polygonal lines considered in the paper, and let $P_n$ be a probability measure on ${\varPi}_n$. Vershik's universality conjectur…

Let a unimodal polygon have two staircase regions with identical sides, possibly differing widths, and let the two boxes delimiting the staircase region be identified as in the pro…