24 problems

Simon’s finiteness conjecture. There are only finitely many knot groups $H$ for which there is an epimorphism $G\twoheadrightarrow H$.

A knot exterior has a $(g,b)$ presentation if its fundamental group has a presentation with $g+b$ generators, $b$ of which are meridional; a knot has a $(g,b)$ decomposition in the…

Lopes–Morales conjecture. Every compact 3-manifold contains such a knot.

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…

Pseudo-meridian conjecture. Knot groups contain infinitely many inequivalent pseudo-meridians.

A generalized torus link group is a group associated with a generalized torus link; in the paper, a group is called flawed according to the definition given in Remark. Flawed-group…

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 $K$ be a knot with group $G(K)$. A pseudo-meridian is an element $g\in G(K)$ that normally generates $G(K)$ but is not the image of a meridian or its inverse under any automorp…

Unknotting conjecture. If a $2$-knot has infinite cyclic knot group, then it is unknotted.

A group is an $R^*$-group if it has no generalized torsion, and it is bi-orderable if it admits a strict total ordering invariant under both left and right multiplication. For a…

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…

Simon’s finite-length conjecture. There is a number $N_G$ such that every epimorphism sequence with $n\geq N_G$ contains an isomorphism.

Let $r\geq 1$ and let $n_1,\ldots,n_r$ be positive integers. A generalized torus knot group is a group with presentation … When $r=2$, these are torus knot groups. Generalized toru…

Let $K$ be a non-trivial knot in $S^3$, let $K(r)$ denote the $3$-manifold obtained by $r$-Dehn surgery on $K$, and let $pi_1(K(r))$ be its fundamental group. Dehn-filling generali…

Silver–Whitten–Williams conjecture. The group of every nontrivial knot has infinitely many pairwise inequivalent killers.

Let $\Gamma_k$ be the knot group and let $R_2(\Gamma_k)$ denote its $\mathrm{SL}(2,\mathbb{C})$ representation space. For a diagonal representation $\rho_\lambda\in R_2(\Gamma_k)$,…

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 $K_{p/q}$ and $K_{p'/q'}$ be 2-bridge knots, and write $K_{p'/q'}\geq K_{p/q}$ when there is a peripheral-structure-preserving epimorphism from the knot group of $K_{p'/q'}$ on…

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…

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…

Let $SK$ and $GK$ denote the square and granny knots, respectively, let $G_n(K)$ be the generalised knot group, and let $H$ be a finite group. Lin–Nelson's homomorphism-counting co…

Let $SK$ and $GK$ denote the square and granny knots, respectively, and let $G_n(K)$ be the generalised knot group associated with a knot $K$. Lin–Nelson's conjecture. The groups ……