6 problems

Satellite-operator dichotomy conjecture. Every satellite operator $P\colon \mathcal{C}\longrightarrow \mathcal{C}$ is constant or has image generating an infinite-rank subgroup of…

Let $\mathcal{C}$ be the smooth knot concordance group. A satellite operator $P$ is rank-expanding along $\{nK\}_{n\in\mathbb{Z}}$ if $\{nK\}_{n\in\mathbb{Z}}$ is a rank-one subgro…

Let $P$ be a pattern, that is, a knot in a solid torus, and let $P$ induce a homomorphism on the smooth or topological knot concordance group. Hedden's conjecture. Then $P$ induces…

Let $Y$ be an integral homology sphere and let $K\in\mathcal{K}(Y)$ be a knot. A knot is local if it lies in a ball in $Y$. Local-knot characterization. The almost-concordance clas…

Let $*$ be either $\operatorname{ex}$ or $\operatorname{top}$, denoting the relevant concordance category, and let $P:\mathcal{C}_*\to\mathcal{C}_*$ be the map induced by a winding…

A winding-number-one operator is a satellite operator $P\colon \mathcal{C}\to\mathcal{C}$ associated to a pattern of winding number $1$, where $\mathcal{C}$ denotes the smooth knot…