10 problems

Genus bound conjecture. For every knot $K$, the displayed inequality holds. The conjecture has been verified for all knots with up to 13 crossings, but remains unproved in general;…

Let $K_1$ and $K_2$ be knots with equal ordinary signatures and equal Arf invariants. Pichelmeyer's conjecture. One should have … This conjecture proposes that the $\mathbb{CP}^2$-…

Murakami–Yasuhara conjecture. There exists a knot $K$ for which equality is achieved:

Minimum-genus Bennequin surface conjecture. Every transverse link $\mathcal{T}$ in $(S^3,\xi_{std})$ is represented by a braid word $w$ whose Bennequin surface $F_w$ contains…

Let $\{(K_n,r_n)\}$ be a twist family of surgeries, where $r_n$ is the corresponding surgery slope on $K_n$. Twist-family surgery finiteness conjecture. For every integer $N\geq 0$…

Let $\{K_n\}$ be a twist family of knots, obtained by twisting along a fixed unknot disjoint from the initial knot, and let $g(K_n)$ denote the genus of $K_n$. Twist-family finiten…

Let $N\geq 0$ be an integer, and let $g(K)$ denote the genus of a knot $K$. Finiteness conjecture for bounded genus. There are only finitely many L-space knots $K$ with…

Let $K$ be an oriented knot. Full $L^2$-Alexander genus conjecture. The full $L^2$-Alexander torsion satisfies … This would make the full $L^2$-Alexander torsion a sharp genus inva…

Let $K$ be a knot. Genus-detecting higher-order torsion conjecture. There exists an epimorphism $\gamma\colon\pi_K\to\Gamma$ onto a torsion-free elementary-amenable group such that…

Let $L(p,q)$ be a lens space with a genus-$1$ Heegaard splitting, and let a simple knot be either the unknot or the union of two arcs contained in meridian disks of the two solid t…