14 problems

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…

Polynomial-time conjecture. For each fixed link $L$, there is a polynomial-time algorithm that decides, for every input link diagram $D$, whether $D\leadsto L$.

Matched-diagram conjecture. (i) Not every link has a matched diagram. (ii) Every oriented link is $t_3$-move equivalent to a link with a matched diagram.

Let $L$ be a link in $S^3$ with a diagram that is prime, twist-reduced, and $3$-highly twisted, and that has at least two twist regions. Hyperbolicity conjecture. The link $L$ is h…

Let $L_1$ and $L_2$ be links, and let $c(L)$ denote the crossing number of a link $L$. Their connected sum is a link obtained by joining $L_1$ and $L_2$ along arcs. Crossing-number…

Let $L_0$ be a prime link, and let $L_n$ be an $m$-string alternating or tangle link constructed from $L_0$ for $m\geq 2$. A link is prime if it cannot be expressed as a nontrivial…

Let $L_0$ be a connected, irreducible link, and let $TL_n$ be an $m$-string tangle link constructed from $L_0$ for $m\geq 2$. A link is non-alternating if it does not admit an alte…

Let $L$ be a $2$-connected link with $n$ crossings, where $n\ge 3$, and define … An Omega1 move, an Omega2 move, and a short pass replacement are the corresponding diagrammatic ope…

Let $L$ be a $2$-connected link with $n$ crossings, where $n\ge 3$. A short and equal pass replacement is the replacement referred to in the source. Crossing-number conjecture. If…

Almost-alternating conjecture. All Turaev genus one links are almost alternating.

Let $D(K)$ be a semi-adequate diagram of a satellite link $K$. Visible satellite torus conjecture. Either $D(K)$ or its mirror image can be obtained using the construction describe…

Let $D(K)$ be a connected, prime, semi-adequate diagram of a link $K$. Primality conjecture. The diagram $D(K)$ must represent a prime link $K$. This would remove the extra hypothe…

Flyping-circuit inequality. For any link diagram $D$,

Twist-number lower-bound conjecture. For every such link and diagram,