7 problems

Hedden–Pinzón-Caicedo conjecture. The only homomorphisms induced by satellite operations are the following:

Let $P$ be a non-constant satellite operator of winding number zero, and let $K$ be a knot. Write $nK$ for the $n$-fold connected sum when $n\geq 0$ and for the corresponding inver…

Let $\mathcal{C}$ be a smooth or topological knot concordance group, and let $P:\mathcal{C}\to\mathcal{C}$ be a satellite operator. Call $P$ non-constant if its image is not a sing…

Let $\mathcal{C}$ denote a smooth or topological knot concordance group, and let a satellite operator be the self-map $P:\mathcal{C}\to\mathcal{C}$ induced by a knot pattern…

Robust-operator infinitude conjecture. The number of distinct robust operators, as $R$ and $\alpha$ vary, is infinite, with the parameter space containing natural independent param…

Whitehead-doubling injectivity conjecture. The Whitehead-doubling operator is injective on $\cal{C}$. Whitehead doubling is presented as a particular satellite operation covered by…

Satellite injectivity conjecture. Many of these operators $R_\alpha$ are injective. This proposes that many self-similarity structures from knots descend to knot concordance; the p…