13 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…

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 $\mathcal{K}$ be the fibered positive knot shown in the paper's counterexample diagram. The unknotting-number conjecture for $\mathcal{K}$. Its unknotting number is … If true,…

Let $K$ and $K'$ be knots in $S^3$, and write $K'\leq K$ when there is a ribbon concordance from $K$ to $K'$. A knot $K$ is positive if it admits an oriented diagram in which every…

Let $K$ be a positive knot. Write $u(K)$ for its unknotting number and $tr(K)$ for its trivializing number. Hanaki's conjecture. … The trivializing number is a knot invariant relat…

Let a positive knot be a knot admitting a diagram with only positive crossings, and let its signature be fixed. Finite-genus conjecture. Positive knots with a given signature shoul…