11 problems

Linked-pair lower-bound conjecture. One has

Positive twist-knot minimization conjecture. Positive twist knot diagrams minimize $v_3$ among all connected irreducible positive diagrams with odd crossing number.

Let $D$ be a positive bireduced knot diagram, and let $lk(D)$ denote the number of linked pairs in $D$. Let $v_2$ be the second Vassiliev invariant. The linked-pair lower-bound con…

Positive-diagram crossing-number conjecture.

Let $K$ be a prime fibered positive knot of genus $g$, and let $\widehat{HFK}_{-1}(K,g-1)$ denote its knot Floer homology in Alexander grading $g-1$ and Maslov grading $-1$, with c…

Let $K$ be a knot, let $\overline{X}$ be the infinite cyclic cover of its exterior, and consider the associated Seifert-form data over $\mathbb{Q}$. A knot is $\mathbb{Q}$-anisotro…

A smooth concordance class is an equivalence class of knots under smooth concordance. A knot is minimal with respect to ribbon concordance if no distinct knot lies below it in the…

Let $K_0,K_1\subset S^3$ be knots, and write $K_0\leq K_1$ when there is a ribbon concordance from $K_1$ to $K_0$. A knot is positive if it admits a diagram in which all crossings…

Perturbed Conway invariant positivity conjecture. If $K\subset S^3$ is a positive knot, then all coefficients of $\delta_1(K)$ are non-positive.

For each integer $n\geq 1$, let $\mathcal{K}_n$ be the knot represented by the positive diagram $D_n$ in the paper's infinite family. The family unknotting-number conjecture. … The…

Let $K$ be a positive fibered knot. Its unknotting number is denoted by $u(K)$, and its 3-genus by $g(K)$. Stoimenow's conjecture. … The conjecture extends the equality known for b…