18 problems

Let $K$ be a Montesinos knot with three branches. Here $M(0; (p,r),(q,s),(2mn\pm 1, 2n^2))$ denotes the Montesinos knot with the displayed parameters. Torisu's conjecture. The knot…

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

Band-sum unknotting-number conjecture. Any band sum of two nontrivial knots has unknotting number greater than one.

Anti-conjecture to the Unknotting Additivity Conjecture. For every non-trivial knot $K$, there is a symbiont knot $K^\prime$ for which

Let $K\#K'$ be a connected sum of knots, let a collection of unknotting crossing arcs for $K\#K'$ be given, and let the 2-sphere specifying the connected sum be fixed. Crossing-arc…

For each integer $n\geq 1$, let $\mathcal{K}_n$ be the knot represented by the positive diagram $D_n$ in the paper's infinite family. The family unknotting-number conjecture. … The…

Let $K$ be a positive fibered knot. Its unknotting number is denoted by $u(K)$, and its 3-genus by $g(K)$. Stoimenow's conjecture. … The conjecture extends the equality known for b…

Delta move finite type invariant conjecture. For some $n$, we have

Winding-number lower-bound conjecture.

Bernhard–Jablan conjecture. Every knot $K$ possesses a minimal-crossing-number diagram $D$ and a crossing in $D$ such that changing the crossing results in a diagram $D'$ for a kno…

Let $P(a_1,\dots,a_n)$ be the pretzel knot with $n$ odd and positive odd integers $a_1,\dots,a_n$, ordered so that $a_1\leq a_2\leq\dots\leq a_n$. Jablan–Sazdanović conjecture. … T…

Uniqueness conjecture. If $D$ contains more than one unknotting crossing of the same sign, then $K$ is a clasp knot.

Kohn's conjecture. If

Torisu's conjecture. The knot satisfies $u(K)=1$ if and only if

Let $T(2,k)$ be the $(2,k)$-torus link, let $T(2,-k)$ be its mirror image, and let $u(L)$ denote the unknotting number of a link $L$. For an odd integer $k$, consider the connected…

Evenness conjecture. For every virtual shadow $S$, $\text{tr}(S)$ is even.

Let $K\subset S^3$ be an alternating knot, and let an alternating diagram be a diagram in which over- and under-crossings alternate along each component. An unknotting crossing is…

Let $u,v$ be parameters and let $L$ be a link for which the supersignature $\sigma_{u,v}(L)$ exists and is finite. Let $u(L)$ denote the unknotting number, and let $T_n$ be the tri…