8 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…

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

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…

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.