56 problems

Hoste's conjecture. Then $\Re(t) > -1$. This conjecture concerns the location of zeros of Alexander polynomials for alternating knots. The source recalls it while analyzing zeros i…

Let $K$ be a knot. Its ropelength $R(K)$ is the minimum length among all representatives of the ambient isotopy class of $K$ with unit thickness, and let $Cr(K)$ denote its crossin…

Let $K$ be a knot, let $c(K)$ denote its crossing number, and let $W(K)$ be its Whitehead double. Tripp's conjecture. The crossing number of any knot coincides with the canonical g…

Let $K$ be a prime alternating knot, and let $K'$ be a 2-twin of $K$, meaning a knot sharing the same two-fold branched covering as $K$. Greene's conjecture. The knot $K'$ must be…

A knot $K$ in $S^3$ is alternating if it admits a diagram in which over- and under-crossings alternate along the knot. Let $K$ be a non-torus fibered alternating knot, let $E(K)$ b…

Let $K$ be a prime alternating knot, and let $D$ be an alternating diagram of $K$ without nugatory crossings, so that $D$ is a minimal diagram. Let $X^{(2)}_K$ denote the double co…

Kauffman–Jablan conjecture. If $K$ has no minimal achiral projection, then $K$ is arborescent.

Let $p$ be a prime, and let a knot have a reduced alternating diagram with determinant $p$. A non-trivial $p$-coloring is a labeling of the diagram's arcs by elements of…

Non-triviality conjecture. $K$ is not trivial for $F$.

Achiral-generator conjecture. Then at least one of the following holds: $K$ is achiral; $K$ is an iterated mutant of its obverse; or $K$ has self-conjugate HOMFLY and/or Kauffman p…

Let $L$ be a prime alternating knot, and let $s(L)$ and ${\text{\it Kh}}'(L)$ be the integer and Laurent polynomial from Khovanov's structural conjecture. The signature of $L$ is t…

Let $n$ be an odd natural number. The determinant characterization conjecture for prime alternating achiral knots. Then $n$ is the determinant of a prime alternating achiral knot i…

Let $\alpha'_p$ be the number of closed prime self-intersecting curves with $p$ crossings, and let $\alpha”_p$ be the number of prime alternating knots with $p$ crossings. Let…

Defect proportion comparison conjecture. For all even $c\geq 4$ and all odd $c\geq 11$, the proportion of $c$-crossing prime alternating links $L$ with $\gamma(L)>\Gamma(L)$ is lar…

Monotonicity conjecture. The following assertions hold:

Median and mode conjecture. For any $c\geq 12$, the medians and modes for $\gamma$ and $\Gamma$ among all $c$-crossing prime alternating knots equal

Let $K$ be an alternating non-trivial $2$-adjacent knot, and let a minimal diagram mean a diagram of $K$ with the minimum crossing number. A $2$-adjacency set is a pair of crossing…

Let $K$ be an alternating knot. A $2$-adjacency set is a pair of crossings exhibiting $2$-adjacency, and the crossings have the same crossing sign when their signs agree. Same-sign…

Ishii's conjecture. The polynomial is alternating: $a_{ij}a_{i'j'}\geq 0$ if $i+j-i'-j'$ is even, and $a_{ij}a_{i'j'}\leq 0$ otherwise. Ishii verified the conjecture for all prime…

Let $Th(p,q)$ denote a Turk’s head knot, with $p\geq 5$ and $q\geq 4$. A knot is super-Kauffman if it admits a minimal alternating diagram whose every crossing has the Kauffman pro…

Minimal ascending number conjecture. One has

Divisibility conjecture. If $K$ has the strong roller-coaster property, then the crossing number of $K$ is divisible by four.

Let $K$notin $\mathcal{D}$ be a knot admitting positive alternating surgeries. Write $\mathcal{S}_{\rm alt}(K)$ for the set of alternating surgery slopes, and let $N$ denote the re…

Let $K$ be a knot, let $\overline{K}$ denote its mirror image, let $r(K)$ denote the minimum ribbon number of a ribbon disk for $K$, and let $c(K)$ denote the crossing number of…

Let $K$ and $K'$ be knots in $S^3$, and write $K'\leq K$ when there is a ribbon concordance from $K$ to $K'$. A knot $K$ is ribbon concordance minimal if $K'\leq K$ implies that…