22 problems

Murakami–Przytycki's conjecture. For every positive fibered knot $K$,

Let … be the Dean knot represented by the closed $p$-braid … where $p>r>1$ and $q,s$ are nonzero integers with $\gcd(q,p)=1$. Let $\Delta_K$ be the Alexander polynomial of $K$. Hir…

Twisted torus knot fiberedness conjecture. A twisted torus knot is fibered if and only if its Alexander polynomial is monic.

Let $K$ be a fibered knot, let $\genusK$ be its genus, let $\hopfK$ be its Hopf invariant, and let $F_K$ have leading term determined by the exponent $\el(K)$. Fibered-knot leading…

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 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 $K$ be a knot, let $\genusK$ denote its genus, let $\hopfK$ denote its Hopf invariant, let $\el(K)$ denote the integer-valued exponent invariant in the leading term of $F_K$, l…

Fibered-knot coefficient conjecture. This coefficient is an integer multiple of

Let $K$ be a prime fibered positive knot of genus $g$, and let $\widehat{HFK}_{-1}(K,g-1)$ denote its knot Floer homology in Alexander grading $g-1$ and Maslov grading $-1$, with c…

Let $K$ be a fibered knot in a manifold $M$, and let $\xi$ be a contact structure on $M$. Non-thickenable-tori conjecture. The knot type $K$ admits non-thickenable tori in $\xi$ if…

Let $K$ be a fibered knot in $S^3$ with irreducible monodromy, and suppose that $K$ does not support the standard contact structure $\xi_{std}$. Irreducible-monodromy conjecture. T…

Let $K$ be a fibered knot, and let $\xi_K$ be the contact structure it supports in the sense of Giroux. Positive Thurston–Bennequin conjecture. In $\xi_K$, the knot type $K$ has Le…

Let $K$ be a hyperbolic knot in $S^3$, let $E_K=S^3\setminus\nu(K)$ be its exterior, and let $x(K)$ denote its Thurston norm, equivalently … Let $\rho^{\mathrm{geom}}$ be a lift of…

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…

Let $K\subset Y$ be a fibered knot of genus $g$, with monodromy $\varphi$, and let $T_\varphi$ denote its mapping torus. Write $\overline{Y}$ and $\overline{K}$ for the orientation…

Schleimer–Thompson conjecture. For any three-manifold $M$, there is a constant $t(M)$ such that if $K\subset M$ is a fibered knot, then the monodromy of $K$ has complexity at most…

Let $K_{0}$ and $K_{1}$ be fibered knots in $S^{3}$ supporting the tight contact structure. Baker's conjecture. If $K_{0}$ and $K_{1}$ are concordant, then $K_{0}=K_{1}$. The conje…

Schleimer's conjecture. There is a constant $t(M)$ such that the monodromy of every fibered knot $K\subset M$ has translation distance in the arc complex of the fiber at most…

Let $K\subset S^3$ be a fibered knot, and let $\xi_K$ be the contact structure induced by an open-book decomposition of $K$. Let $\xi_{Std}$ denote the standard contact structure o…

Let $K$ be a fibered oriented knot with monodromy $f$, and normalize its full $L^2$-Alexander torsion so that $\tau^{(2)}(K)(1)=1$. Full $L^2$-Alexander dilatation conjecture. Then…

Periodic-representation detection conjecture. The knot $k$ is nonfibered if and only if there exist an integer $N>0$ and a periodic representation $\rho\in\Phi_{S_N}({\cal G})$ suc…

Let $k$ be a knot, let $cpi$ be its knot group, and for each finite group ${\Sigma}$ let $h_{\Sigma}$ denote the topological entropy of the representation shift associated with…