19 problems

Let $K\subset S^3$ be a knot that is not $\gamma$--primitive, and let a minimal genus Heegaard splitting of its exterior $E(K)$ be given. A boundary stabilization is the genus-incr…

Let $K$ be a knot in $S^3$. Suppose there exists a rational number $r$ such that the $3$-manifold obtained by $r$-surgery on $K$ is homeomorphic to the Poincaré homology sphere…

Let $K$ be a strongly quasinegative knot. Assume that its stratified sum converges to a power series $F_K\in\mathop{\mathrm{FKring}}$, and let $\Phi_K(x,q)$ denote the stability se…

Let $K$ be a nice knot, let $F_K(x,q)$ be its $F_K$ series, let $N_p(K,x)$ be its ADO polynomial, let $\Delta_K(x)$ be its Alexander polynomial, and let $\zeta_p=e^{2\pi i/p}$ be a…

Let $K$ be a strongly quasipositive knot with genus $g$, and let $\widehat{\mathit{HFK}}_m(K,g)$ denote the summand of knot Floer homology in Maslov grading $m$ and top Alexander g…

Let $K$ be a strongly quasinegative knot, and let the colored Jones function denote the sequence of colored Jones polynomials of $K$. A stability series is a Laurent power series i…

Let $K$ be a knot. A knot is nice when it admits a braid representative equipped with an inversion datum, as defined in the paper. It is fibered when its complement fibers over the…

Let $X$ be a homotopy $\mathbb{CP}^2$ constructed in Proposition from a Berge knot. Standardness conjecture. The manifold $X$ is diffeomorphic to the standard $\mathbb{CP}^2$. Thes…

Let $K$ be a null-homologous knot in $\mathbb{RP}^3$. Write $\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)$ for its reduced Khovanov homology over $\mathbb{Z}/2$. Rank-three Khovan…

Let $e$ be an embedding of a pochette into $S^4$, let $S_e$ denote its associated spun 2-knot, and let $S^4(e,p/q,\varepsilon)$ denote the result of pochette surgery with parameter…

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 $K$ be a knot, let $p$ be a fixed prime number, and let $\pi$ be its knot group. For a representation $\rho:\pi\to \operatorname{GL}_N O_{\mathfrak{p}}$, let $\mu_1$ denote the…

Simple-knot detection conjecture. If

Let $K\subset S^3$ be a knot, let $f_K(x,q)$ be its rescaled knot series, let $\widehat A$ be the quantum polynomial of $K$, and let $\Delta_K(x)$ be its Alexander polynomial. Defi…

Let $K$ be a knot in a lens space $L$ and let surgery on $K$ produce a non-prime $3$-manifold $Y$. A knot $K$ is hyperbolic if its exterior admits a complete finite-volume hyperbol…

Let $K$ be a knot in a closed connected orientable $3$--manifold $Y$, with exterior $M=Y\setminus\operatorname{int}N(K)$ irreducible and not homeomorphic to the solid torus. A slop…

For a knot $K$, let $X$ be its complement and let $\mathbb C^{\mathrm{univ}}[\chi(X)]$ denote the universal $SL_2$-character ring of $\pi_1(X)$, equivalently the skein module at…

Finite lead-coefficient conjecture. The lead coefficient of {\mathcal{T}_K^{\vphantom{x}}\hspace{-0.5em}\raisebox{0.8ex}{scriptstyle X0}} has a finite value at $x$.

Let $Q\subset S^3$ be the square knot, and let Property $2R$ mean that whenever $Q$ is a component of a $2$-component link whose integral surgery yields … a sequence of handle slid…