81 problems

Free knots initially appeared as isotopy classes of Gauss words, namely equivalence classes of Gauss words under the relevant Reidemeister moves. Turaev's conjecture. Every free kn…

Let $K$ and $K'$ be virtual knots, and let $\mathcal{B}(K)$ and $\mathcal{B}(K')$ denote their fundamental biquandles. Two virtual knots are identified up to orientation-reversed m…

A virtual three-manifold is a Kirby equivalence class of framed virtual links, and its fundamental group is the combinatorial fundamental group associated with the virtual link. Vi…

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

A virtual knot has an associated fundamental biquandle, an algebraic invariant defined from a virtual-knot diagram, and two virtual knots are identified up to mirror reflection whe…

For an oriented virtual knot $K$, let $\mathcal{B}(K)$ denote its fundamental biquandle, the algebraic structure associated with $K$ that encodes its Reidemeister moves. Fundamenta…

Let $N$ be a fixed $n$-manifold, and let $\Phi_N:V_m\rightarrow V_{m+n}$ be the construction sending $(F\times [0,1],L)$ to $(F\times N\times [0,1],L\times N)$. Here $V_m$ denotes…

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…

A shadow code is the flat Gauss code obtained by replacing the classical crossings of a virtual knot diagram by flat crossings. Kauffman's conjecture. Virtual knots with classical…

A finite-type invariant of a classical knot is an invariant taking values in some target group and having finite Vassiliev degree; a long virtual knot is a virtual knot with fixed…

Virtual torus-knot span conjecture. The non-zero minimum of the product

Kishino span conjecture. The span of Kishino's knot is $4$.

Let $G$ be a graph. Call $G$ intrinsically knotted if every classical spatial embedding of $G$ contains a non-trivial knot, and call it intrinsically virtually knotted of degree 1…

Let $K$ be a classical knot, and let $vu(K)$ denote its virtual unknotting number, the minimum number of crossing virtualizations needed to obtain a trivial virtual knot. Virtual u…

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

Let $K_m'(t)$ be the modified family of virtual knot diagrams illustrated in the paper. Let $\langle K\rangle_N$ denote the $N$-strand bracket polynomial. The modified Family B con…

Let $K_D(t)$ denote the virtual knot diagrams in Family A depicted in the paper's schematic, and let $\langle K\rangle_N$ denote the $N$-strand bracket polynomial. The Family A det…

Let $K$ be a classical knot diagram, and let $\langle K\rangle$ denote its bracket polynomial. The bracket-polynomial unknot-detection conjecture. If … then $K$ is the unknot. For…

Let Family A and Family B be the infinite families of virtual knot diagrams constructed in the paper. The family detection conjecture. The 3-strand bracket polynomial detects all t…

Let $K$ be a virtual knot diagram. The 3-strand bracket polynomial conjecture. The 3-strand bracket polynomial detects all non-trivial virtual knot diagrams. This would establish t…

A 3-strand bracket polynomial is the bracket polynomial constructed using three strands for a virtual knot diagram. Detection conjecture. The 3-strand bracket polynomial detects vi…

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

Let $K_I$ be the virtual knot diagram discovered by Kishino. The paper notes that its generalized Alexander polynomial and Alexander Biquandle are trivial, while its general Biquan…

Single-loop reduction conjecture. A single-loop state with exactly two distinct virtual crossings can be undone by virtual moves.

Let $K_1$ and $K_2$ be representatives of virtual knots $[K_1]$ and $[K_2]$, and choose any connected sum $K_1\#K_2$. Let $upr$ denote Manturov's parity group projection from virtu…