18 problems

Let $T(c)$ be the set of objects with crossing number $c$ considered in the paper, and let the variance of their signatures be computed with respect to the uniform distribution on…

Let $c\geq 3$, and let $n=\left\lceil\frac{c+1}{2}\right\rceil$. For each $b$ with $2\leq b\leq n$, let $k_{c,b}$ denote the number of $2$-bridge knots with crossing number $c$ and…

Let $K=C(p,q)$ be a positive 2-bridge knot with rational number $p/q$, where $p,q>20$. Let $v_3$, $a_2$, $a_4$, $\det(K)$, and $g$ denote the knot invariants used in the obstructio…

Vernitski's conjecture. The knot semigroup of every 2-bridge knot is isomorphic to an alternating sum semigroup.

Low-level 2-bridge knot conjecture. Every $2$-bridge knot of level number at most two has Conway parameters of one of the forms above.

Sliceness conjecture for 2-bridge knots. For all $2$-bridge knots $K$, $g_{4}(K)=0$ if and only if $g_{4}^{{\rm top}}(K)=0$.

Let $K_{m^2,q}$ be a 2-bridge knot, and let $\mathcal{R}$ denote the family of ribbon knots. The Casson–Gordon signature invariants associated to the double branched cover are the…

Let $K$ be a 2-bridge knot, and let $M$ be the module defined in the source by the referenced construction. Let the nonsymmetric skein module of $K$ be regarded as a module over th…

Large-parameter stability conjecture. There exists a positive integer $N_m$ such that: if $m$ is even, then $K(r_m)$ is $c$-stable for $k\geq N_m$; and if $m$ is odd, then $K(r_m)$…

Let $K_1=\Phi(\hat{\bf a})$ and $K_2=\Phi(\hat{\bf b})$ be knots that share a common ORS knot cover, and suppose neither is an ORS cover of the other. Let $\mathcal{S}_{even}$ deno…

Let $K$ be a 2-bridge knot, and suppose that its boundary-slope set has exactly four distinct elements. A 2-bridge knot $K_1$ is minimal with respect to the Silver–Whitten partial…

Let $K_{p/q}$ and $K_{p'/q'}$ be 2-bridge knots, and write $K_{p'/q'}\geq K_{p/q}$ when there is a peripheral-structure-preserving epimorphism from the knot group of $K_{p'/q'}$ on…

Silver--Williams conjecture. Under these assumptions, there is an odd integer $m$ such that

Representation-polynomial divisibility conjecture. If $f_2(z)$ divides $f_1(z)$, then there exists an epimorphism

Generalized evaluation conjecture.

Silver–Williams conjecture.

Mod 4 characterization conjecture. $\mu(r)=-1$ if and only if $r$ is (mod 4)-equivalent to either $[p]$ or $[3p]$.

Let $k$ be a $2$-bridge knot, and let ${\gamma}_\phi$ be a total representation associated with a nonabelian parabolic representation ${\gamma}:\pi\to\operatorname{SL}_2\mathbb C$,…