17 problems

Non-trivial realization conjecture. Then there exists a minimal genus representation resulting in a non-trivial realization.

Virtual crossing realization conjecture. Every maximally virtualized virtual isotopy sequence with switch-free classical initial and final diagrams admits a valid virtual crossing…

Cromwell's conjecture. Then $D$ is composite.

Let $\mathrm{minwr}_{\mathbb{R}^2}$ and $\mathrm{minwr}_{S^2}$ denote the minimum writhe of a max-tb Legendrian unknot front projection that is smoothly hard on the plane and spher…

Two-bridge knot transvergent diagram conjecture. A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings if and only if one of these two conditions is fulfilled.

Infinite virtual connect-sum conjecture. There are infinitely many distinct virtual links that decompose as a connect sum of $V_1$ and $V_2$.

Fix a ribbon knot and consider ribbons spanning it. A ribbon 0-move, ribbon 1-move, finger move, and ribbon 3-move are the ribbon transformations described in the paper's figures;…

Let $K$ be a minimal free knot diagram with at least four crossings that is not a foil or a connected sum of solely foil knots. An assignment of crossings resolves every free cross…

Let $S$ be a reduced shadow with $n$ crossings. Say that $S$ resolves into a knot type if some assignment of overcrossings and undercrossings to its crossings produces a diagram of…

A knot diagram is straight if, after labeling its $n$ crossings in traversal order, its knot word has the form … where $\sigma\in S_n$; a straight number $\mathtt{str}(K)$ is the m…

Minimal-diagram bridge-number realization conjecture. For any knot $K$,

Let $K$ be a virtual knot, and let $D$ and $D'$ be diagrams of $K$. The generalized Reidemeister-move conjecture. There exist two diagrams $D$ and $D'$ such that every sequence of…

Let a knot have weak property PT when it satisfies the property defined in the paper. Positive-density weak property PT conjecture. There exist $\varepsilon>0$ and $N>0$ such that,…

Let a knot diagram have strong property PT when it has the strong property PT defined in the paper. Strong property PT conjecture. Each prime knot has strong property PT. The paper…

Consider a knot or link diagram satisfying the conditions of Theorem; in particular, the diagram is a sufficiently long plat with at least three crossings in each twist region. Min…

Let $K$ be a framed four-valent graph, viewed as a free knot diagram, and let its girth be the length of its shortest cycle. If … then the unique minimal diagram conjecture. The di…

A wave move is a local move on a knot diagram that may preserve or reduce its crossing number. Wave-move conjecture. Crossing-number-preserving and crossing-number-reducing wave mo…