33 problems

Let $K$ be a knot, let $u(K)$ denote its unknotting number, and let $D$ be a minimum crossing number projection of $K$. A crossing change in $D$ produces a diagram $D^\prime$ of a…

Let a knot $P$ have strict BJ-property if altering one crossing in a minimal diagram of $P$ produces a knot $J$ with $u(J)<u(P)$, where $u$ is unknotting number. Kohn's conjecture.…

Stoimenow's conjecture.

A knot diagram's unknotting number $u(K)$ is the minimum number of crossing changes, over all diagrams of a knot $K$, needed to transform $K$ into the unknot. For coprime positive…

Unknotting-number question. Is it true that

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

A rational knot is a knot represented by a rational diagram, equivalently by an iterated fraction. An even-integer expression is an iterated-fraction expression in which all intege…

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 $\includegraphics{mm.pdf}_{\ell}$ denote the $\ell$th knot in the Manolescu–Marengon family, and let $\operatorname{HFK}^-(\includegraphics{mm.pdf}_{\ell})$ be its minus knot F…

For a knot $K$, let $u(K)$ be the minimal number of crossing changes needed to transform $K$ into the unknot. For knots $K$ and $J$, their connected sum is denoted by $K\#J$. Unkno…

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

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…

Let $K$ be a knot. Write $u(K)$ for its unknotting number, and let $u^{c}(K)$ denote the minimum number of crossing changes needed to unknot $K$ through diagrams whose crossing num…

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 $\mathcal{K}$ be the fibered positive knot shown in the paper's counterexample diagram. The unknotting-number conjecture for $\mathcal{K}$. Its unknotting number is … If true,…

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

Winding-number lower-bound conjecture.

Let $u(K)$ denote the unknotting number of a knot $K$. A knot $P$ is strictly U-regular if … whenever $P$ is a factor of a knot $K\neq P$. Strict U-regularity conjecture. All knots…

Unknotting-number conjecture. The authors conjecture that

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.

Let $K$subset S^3 be a knot. Denote by $n(K)$ the minimal size of a hermitian matrix over $\mathbb{Z}[t^{\pm 1}]$ representing the Blanchfield pairing of $K$ whose specialization a…

Pretzel-family conjecture. The assertion proved for the $p+q=-2$ family is also true for the remaining $p+q=2,4$ cases.

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