9 problems

Two-bridge framing conjecture. For $m,l\in\mathbb{Z}^+$, the natural framing of this knot is

Casson–Gordon's conjecture. Every 2-bridge ribbon knot is contained in one of the three families of Casson and Gordon.

Let $K$ be a two-bridge knot. Call $K$ $\mathcal H$-thin if all generators of its triply graded homology $\mathcal H(K)$ have the same $\delta$-grading, and let $S(K)$ denote the a…

Lamm's conjecture. $c_t(K)=c(K)$ if and only if $K$ can be represented by a continued fraction expansion $[a_1,\ldots,a_n]$ of type (A) or type (B).

Two-bridge knot transvergent diagram conjecture. A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings if and only if one of these two conditions is fulfilled.

Dunfield's conjecture. For every rational slope $r\in I$, Dehn filling $K$ along slope $r$ is orderable.

Let … be the Alexander polynomial of a two-bridge knot. Convex-concave conjecture. There exists an integer $k\leq n$ such that $(a_0,\ldots,a_k)$ is convex and $(a_k,\ldots,a_n)$ i…

Let $p$ be an odd prime, let $k$ be a primitive $(p-1)$-st root of $1$ modulo $p$, and suppose that $K(r)$ is a $2$-bridge knot with $\Delta_{K(r)}(k)\equiv0\pmod p$. Let…

Let $K$ be a knot, and let $\pi_1(S^3\smallsetminus K)$ denote its knot group. Suppose this group is generated by two conjugate elements. Reid's conjecture. The elements are periph…