11 problems

J. Simon's finiteness conjecture. There are only finitely many knots $k_i \in {\mathbb S}^3$ for which such an epimorphism exists.

Let $\mathcal{T}ang$ be the category of tangles, let $\Lambda\mathbf{Grp}$ be the category of groups, and let $\Gamma\colon\mathcal{T}ang\to\Lambda\mathbf{Grp}$ be the tangle group…

Let $G_n(SK)$ and $G_n(GK)$ denote the generalized knot groups associated with the square knot $SK$ and granny knot $GK$, respectively, for integers $n\geq 2$. Non-isomorphism conj…

A rat-group is obtained by deleting the final relator from an Artin $n$-presentation of the trivial group, so it has a deficiency-one presentation. A knot group is characterized by…

Let $J$ and $K$ be knots with knot groups $\pi_J$ and $\pi_K$, and let $\varphi:\pi_J\twoheadrightarrow\pi_K$ be a surjective homomorphism. Simon’s genus monotonicity conjecture. T…

Let $G$ be the fundamental group of a $3$-manifold. A non-trivial element $g\in G$ is a generalized torsion element if some non-empty finite product of conjugates of $g$ equals the…

Meridional rank versus bridge number conjecture. The minimum number of meridional generators of $\pi_1(E(K))$ is equal to

Let $\mathcal{K}$ be a knotted surface in $S^4$, and let $S^4\setminus\mathcal{K}$ be its exterior. The fundamental group of the exterior is $\pi_1(S^4\setminus\mathcal{K})$. Cycli…

Let $X$ be a knot exterior in a closed orientable $3$-manifold such that $H_1(X;\mathbf{Q})\cong\mathbf{Q}$. Simon-type knot-exterior conjecture. Then $\pi_1(X)$ surjects onto only…

A knot group is the fundamental group of the complement of a knot in $S^3$. Simon's conjecture. A knot group can surject onto only finitely many other knot groups. This is an attri…

Let $K$ be a knot, and let $\pi_1(S^3\smallsetminus K)$ denote its knot group. Suppose this group is generated by two conjugate elements. Reid's conjecture. The elements are periph…