37 problems

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…

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

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 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 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…

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…

Let $K$ be an alternating knot in $S^{3}$ that is non-amphicheiral and is not a $(2,p)$-torus knot. Alternating-knot nonexistence conjecture. The knot $K$ does not admit chirally c…

Let $K$ be an alternating knot with normalized Alexander polynomial … where $g$ is the genus of $K$. Fox Trapezoidal Conjecture. The coefficients satisfy … Moreover, if…

Tie-knot conjecture. All tie knots are alternating 2-bridge knots.

Let $K\subset S^3$ be a knot, let $p_K$ denote its Jones period, and let $d_+[J_K(n)]$ and $d_-[J_K(n)]$ denote the maximal and minimal degrees of its colored Jones polynomial in…

Let $K$ be an alternating knot, and let $c(K)$ denote its crossing number and $\mathtt{str}(K)$ its straight number. Suppose … where $n\geq 1$, and let $K'$ be the knot obtained by…

Torus-knot alternating-concordance conjecture. A sum of torus knots is concordant to an alternating knot if and only if it is a sum of $(2n+1,2)$ torus knots.

Let $K$ be an alternating knot with signature satisfying $|\sigma(K)|=2k$, and let its Alexander polynomial be … where $a_j>0$. Hirasawa–Murasugi conjecture. The coefficients satis…

Let $K$ be a hyperbolic alternating knot. Here, $\operatorname{vol}(K)$ denotes the hyperbolic volume of the complement $S^3\backslash K$, and $\det(K)$ denotes the determinant of…

Alternating-knot coil number conjecture. For an alternating knot $K$, it holds that