15 problems

Let $K$ be a slice knot in $S^3$, and let $F$ be a genus one Seifert surface for $K$. A slice derivative for $F$ is a simple closed curve on $F$ that is homologically nontrivial, h…

A link diagram determines a Seifert surface by applying Seifert's algorithm. The resulting surface is minimal genus when it has the smallest genus among all Seifert surfaces for th…

Let $F\subset S^3$ be a quasipositive Seifert surface. A proper arc in $F$ is an arc with endpoints on $\partial F$ and interior in the interior of $F$. Let $A(O,-1)$ be the surfac…

Morton's conjecture. For every Seifert surface $S(L)$ of $L$,

For a link $L$, let $R$ be a Seifert surface and let $h(R)$ denote its handle number. Define … Let $R_1\natural R_2$ denote the plumbing of Seifert surfaces. Przytycki–Rudolph–Wada…

Subadditivity conjecture. The Morse–Novikov number should satisfy

Let $K$) and $L$ be prime knots, and let $S(K)$ and $S(L)$ denote their hosting sets. Hosting-set rigidity conjecture. If … then $K=L$. This asks whether a prime knot is determined…

Minimal-genus conjecture. Minimal genus is achieved when $|L_E|$ is minimal.

Let $K$ be a knot, let $n\in \frac{1}{2}\mathbb{N}_0$, and let an $(n)$-solvable derivative mean a derivative of $K$ that is $(n)$-solvable. Solvable Kauffman conjecture. For every…

Let $K$ be a knot, and let a slice derivative mean a derivative of $K$ that is slice in the relevant category. Generalised Kauffman conjecture. If $K$ is a topologically (respectiv…

Higher-genus slice derivative conjecture. There exists a link

Kauffman's conjecture. There exists an essential simple closed curve $d$ on $F$ such that

Let $K$ admit a genus-one Seifert surface and a derivative $J$; consequently, $J$ is a knot and … Suppose $m\notin\{0,-1\}$. Genus-one derivative conjecture. The algebraic concorda…

Let $K$ be a slice knot. A Seifert surface $F$ stabilizes to a Seifert surface $F'$ if $F'$ is obtained from $F$ by finitely many trivial one-handle attachments. A Levine surface i…

Let $K$ be a slice knot, and let $F$ be a genus-one Seifert surface for $K$. For a simple closed curve $d$ on $F$, write $d^+$ for its positive push-off and…