3 problems

Let $K$ be a knot in $S^3$. An essential tangle decomposition is an essential $n$–string tangle decomposition for some integer $n$, meaning that there is a $2$–sphere intersecting…

Let $K$ be a knot in a $3$--manifold $Y$. An essential $n$--string tangle decomposition is an embedded sphere $Q$ intersecting $K$ transversally in $2n$ points such that…

Let $K$ be a knot in $S^3$. An essential Conway sphere is an embedded sphere intersecting $K$ transversally in four points such that the resulting four-punctured sphere is incompre…