14 problems

Let $k$ be a fixed number, and consider links chosen randomly from links represented by grids whose size tends to infinity. Vanishing fixed-linking-number conjecture. The probabili…

Ng–Vértesi's general grid-size conjecture. The representative $\Lambda$ can be represented on a grid of size $\alpha(K)+m$.

Ng–Vértesi's conjecture. A Legendrian knot of maximal Thurston–Bennequin number has a grid diagram of minimum grid number; equivalently, it can be represented by a grid diagram of…

Let $w$ denote the writhe of a knot represented by an $n\times n$ grid diagram. The distribution is considered for knots encoded by two $n$-permutations recording the order of the…

Let $K\subset L(p,q)$ be a knot and let $m(K)\subset L(p,p-q)$ denote its mirror. Consider Legendrian representatives of these knots in the universally tight contact structures on…

Grid-diagram agreement conjecture. The maps described in terms of grid diagrams by Sarkar and Ozsváth–Stipsicz–Szabó are equal to the maps defined in this paper for elementary link…

Grid-cobordism functoriality conjecture. Grid diagram maps induce maps on $\operatorname{HFK}^-$ that are invariant with respect to marked smooth isotopy classes of surfaces.

Let $G$ be a grid diagram, and let a formal complex structure on $G$ be a combinatorial count of domains on the grid satisfying the constraints imposed by positivity of intersectio…

A grid diagram represents a Legendrian knot, with its grid size measuring the complexity of that representative. For a topological knot $K$, let $\overline{tb}(K)$ denote its maxim…

Let $T_{p,2}$ be a torus knot, let $K_L$ be a Legendrian knot of topological type $T_{p,2}$, and let $\alpha(T_{p,2})$ denote the grid number of $T_{p,2}$. Consider a grid diagram…

Positive-pair vanishing conjecture.

Enhanced-domain vanishing conjecture.

Simple knot Floer homology conjecture. If

Berge conjecture. If surgery on $K\subset L(p,q)$ yields the three-sphere, then $K$ has grid number one.