30 problems

A tight fibered knot is a fibered knot supporting the tight contact structure. A knot is prime if it cannot be expressed as a nontrivial connected sum. The smooth knot concordance…

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…

Stoimenow's conjecture.

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

Let $K(N,p,q)$ be a simple minimal knot, where $N$, $p$, and $q$ are the parameters in its standard construction. The knot is fibered if its complement $S^3\setminus K(N,p,q)$ fibe…

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…

Let $L(\alpha,\beta)$ be a lens space, and consider its null-homologous fibered knots. The minimal genus is the least genus among such knots in $L(\alpha,\beta)$. Existence conject…

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

Let $K$ be a strongly quasinegative knot, let $\genusK$ denote its genus, and let $\el(K)$ denote the exponent in the leading coefficient of its $F_K$ series. Strongly quasinegativ…

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…

Ghiggini–Spano's conjecture. More generally than the established identification

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…

Unknotting-number conjecture for fibered positive knots. For every fibered positive knot $K$, one has

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…

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…

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…

Linear independence conjecture. Tight fibered knots are linearly independent in the concordance group.