47 problems

Virtual Legendrian isotopy conjecture. Two Legendrian links in $ST^*S^m$ or $ST^*\mathbb C P^n$ are Legendrian isotopic if and only if they are virtually Legendrian isotopic. The s…

Strong virtual Legendrian unlinking conjecture. There is no non-negative strong allowable virtual Legendrian isotopy of one component of $L$ to the other.

Strong virtual Legendrian isotopy conjecture. There exists no strong allowable positive, or non-constant non-negative, virtual Legendrian isotopy of the sphere fiber of $ST^*M$ to…

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…

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 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…

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

Let $K_1$ 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…

Let $K$ be a virtual knot. Denote by $da^{(u)}(K)$ its unrestricted danceability index and by $b_1(K)$ its virtual bridge index. The unrestricted danceability–bridge index conjectu…

Vanishing conjecture. For all $m\geq 1$, if $K$ is a slice virtual knot, then

Negative stabilization nonuniqueness conjecture. Negative stabilization is not a unique operation on virtual transverse knots. That is, there are two ways to negatively stabilize s…

Virtual transverse triviality conjecture. Then $x$ and $y$ represent the same element of $U_n(VTK)$ for all $n$.

Transverse-to-virtual injectivity conjecture. No two transverse knots are the same virtual transverse knot.

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

Infinite-volume composition conjecture. There exists a sequence of compositions $W_i$ such that

Let $\mathscr{VC}$ be the concordance group of long virtual knots. The almost classical knot $6.85091$ is the closure of a long virtual knot $\upsilon$. The order-two non-classical…

Let $L_1$ and $L_2$ be virtual knots or links, and let virtual mosaics represent them. Two virtual mosaics are equivalent when they represent equivalent virtual knots or links. Vir…

Let $K$ be a virtual knot of genus one, and let a torus representation be a representation of $K$ by a knot diagram in the thickened torus. The surface bracket states of such a rep…

A finite type invariant of classical knots is an invariant filtered by finite-dimensional vector spaces. A finite type invariant of long virtual knots is defined analogously in the…

Dye–Kaestner–Kauffman conjecture. For every classical knot $K$,

Vanishing conjecture. If $K$ is slice, then

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…

Classical knotoid embedding conjecture. If two knotoid diagrams in $S^2$ are virtually equivalent, then they are equivalent to each other by finitely many $\Omega$-moves and isotop…