29 problems

Single-component reduction conjecture. Any single-component diagram decorated with no more than two distinct virtual crossings reduces by detour moves to a circle.

Let $D$ be the virtual knot diagram in Figure 21, and let $E$ be its shadow, obtained by replacing the classical crossings of $D$ by flat crossings while keeping flat crossings dis…

Murasugi–Przytycki's conjecture. For every alternating diagram $D$, this inequality is exact:

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

A Vassiliev invariant is a knot invariant extended to singular knots by resolving each double point as the difference between a positive and a negative crossing; an invariant is of…

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…

The $6_3$ transvergent diagram conjecture. The knot $6_3$ has no transvergent diagram with $6$ crossings.

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 $V_2$.

Let $D$ be a crossing-number-minimizing diagram of a prime knot $K$ with at most 16 crossings. Write $\omega(D)$ for the Wirtinger number of $D$ and $\omega(K)$ for the Wirtinger n…

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;…

Generalised Tsukamoto conjecture. For every natural number $k$, there exists a finite set $\mathcal{U}_k$ of generalised Tsukamoto moves such that every $k$-almost alternating diag…

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 $K$,

Fertile-knot finiteness conjecture. There are no fertile knots with crossing number greater than seven.

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…

A reduced alternating diagram is an alternating link diagram with no reducible crossings, and a flype is the local move on link diagrams described in the source. Tait–Little flype…

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…

Universal success conjecture. The procedure ${\mathcal P}$ successfully simplifies every link diagram.