26 problems

Let $K\subset S^3$ be a hyperbolic knot, let $g(K)$ denote its genus, and let $\mathcal T_K(t)\in\mathbb C[t^{\pm1}]$ be the normalized hyperbolic torsion polynomial associated wit…

Finiteness conjecture. For every nonfibered knot, there is a curve component of the $SL(2,\mathbb{C})$-character variety containing only finitely many characters whose twisted Alex…

Twisted Alexander polynomial detection conjecture. If, for every epimorphism $\alpha:\pi_1(N)\to G$ onto a finite group, $\Delta^{\alpha}_{N,\phi}$ is monic and

Regular-representation congruence. The twisted Alexander polynomial satisfies

For a representation $\rho:\pi\to {\rm SL}(2,\mathbb{C})$, let the twisted Alexander polynomial (TAP) associated to its conjugacy class be denoted by $\Delta_L^{[\rho]}$. If $K$ is…

Let $M$ be a cusped hyperbolic 3-manifold, let $\mathcal{T}$ be an ideal triangulation, and choose an epimorphism $\alpha:\pi_1(M)\twoheadrightarrow\mathbb{Z}$, equivalently an inf…

Let $K$ be a hyperbolic knot, let $g(K)$ denote its genus, and let $\rho_{\rm holo}:\pi(K)\to\operatorname{SL}(2,\mathbb{C})$ be a lift of the discrete and faithful representation…

Morifuji–Tran conjecture. The polynomial determines the Thurston norm, namely,

Let $C$ be a curve whose components all intersect transversely, let $\rho$ range over the representations under consideration, and let $\Delta_1$ and $\Delta_0$ denote the twisted…

Let $K\subset S^3$ be a hyperbolic oriented knot, and let $\pi_K$ denote its knot group. Let $\alpha\colon \pi_K\to \operatorname{SL}(2,\mathbb{C})$ be a lift of the discrete and f…

Finiteness conjecture. For every nontrivial knot, there is an irreducible component of $X(K)$ satisfying the finiteness condition associated with detecting fiberedness and the genu…

Let $K$ be an oriented knot and let $D_p$ be the dihedral group of order $2p$, with $p=2\ell+1$ an odd prime. Let $\alpha\colon\pi_1(X_K)\to D_p$ be an epimorphism. Hirasawa–Murasu…

Let $K$ be an oriented knot, let $p$ be a prime, and let $n\in\mathbb{N}$. Define the $n$-th cyclotomic polynomial by … Let $A_{p,n}=\mathbb{F}_p[t^{\pm1}]/(\phi_n(t))$, with the a…

Hyperbolic torsion polynomial detection conjecture. The hyperbolic torsion polynomial $4\mathcal{T}_K4$ determines $x(K)$, or equivalently its genus. Moreover, $K$ is fibered if an…

Let $K$ be a nonfibered hyperbolic knot, and let $X_0(K)$ be the canonical component of its character variety, namely the curve containing the character of a discrete faithful repr…

Let $(N,\phi)$ be a manifold pair, where $N$ is a $3$-manifold and $\phi\in H^1(N;\mathbb Z)$ is the associated cohomology class. For an epimorphism $\alpha:\pi_1(N)\to G$ onto a f…

Silver--Williams conjecture. Under these assumptions, there is an odd integer $m$ such that

Morifuji's conjecture. The twisted Alexander polynomial corresponding to a generic' representation should detect whether the manifold is fibered.

Let $K$ be a knot with knot group $G(K)$, and suppose there is a homomorphism $f:G(K)\twoheadrightarrow M(n\mid p,k)$. Let $N=\langle s\rangle=\{1,s,s^2,\ldots,s^{n-1}\}$ and let…

Let $p$ be an odd prime, let $k$ be a primitive $(p-1)$-st root of $1$ modulo $p$, and suppose that $K(r)$ is a $2$-bridge knot with $\Delta_{K(r)}(k)\equiv0\pmod p$. Let…

Let $D_p$ be a dihedral group of order $2p$, where $p=2n+1$ is prime. Let $\widehat{\rho}:G(K)\to D_p\to GL(p,\mathbb{Z})$ be a non-abelian representation of the knot group of a kn…

Let $K$ be a knot whose knot group $G(K)$ maps onto the non-abelian semidirect product … where $p$ is an odd prime and $k$ is a primitive $m$-th root of $1$ modulo $p$. Let…

Generalized evaluation conjecture.

Silver–Williams conjecture.

Let $k$ be a knot of genus $g$, let $\gamma$ be a finite-image representation of its knot group into permutation matrices of degree $n$, and let ${\cal O}_{\gamma}$ be the number o…