8 problems

Habiro's conjecture. Two links in $M$ are equivalent if and only if they are $C_{\infty}^M$-equivalent.

Knot and link replacement faithfulness conjecture. $X_K$ is diffeomorphic to $X_{K'}$ if and only if $K$ is isotopic to $K'$.

Let $\mathcal{T}$ be a rainbow diagram of an $r$-component unlink of $2$-spheres, and let $\mathcal{U}_r$ be the crossingless rainbow diagram with $r$ strands. MTWS conjecture. The…

Let two rainbow diagrams represent isotopic surfaces. A rainbow moves conjecture. Any two rainbow diagrams representing isotopic surfaces are related by a sequence of rainbow moves…

Let $L$ be a link constructed in a solid torus $\mathcal{T}$ by the knotted ribbon-link construction described in the source. Knotted-ribbon hyperbolicity conjecture. A given link…

Let $M$ be a closed orientable $3$-manifold. A TGS link is a link $K$ embedded in a $3$-manifold such that $M\setminus K$ is hyperbolic and $K$ has at least one totally geodesic sp…

Positive-diagram conjecture.

Vassiliev conjecture for $N$. If all finite-type invariants of $L_1^N$ and $L_2^N$ agree, then