Abstract: Constant cycle curves on a K3 surface $X$ defined over $\mathbb{C}$ are curves whose points all define the Beauville-Voisin class in the Chow group of $X$. They were first considered by Huybrechts and Voisin as a generalization of the notion of rational curve. In this talk, we introduce correspondences $Z \subseteq X\times X$ acting on constant cycle curves, and we study geometric examples that can potentially improve our current understanding of constant cycle curves. More precisely, for a general primitively polarised K3 surface $(X, H)$ of genus $p\geq 2$, we consider for any $k\geq 2$ and any $0\leq \delta\leq p$ the locus $Z_{k,\delta}(X,H)\subseteq X\times X$ of pairs of points $(p,q)$ contained in some $\delta$-nodal curve $C$ with the property that $p-q$ is $k$-torsion in the Jacobian of the normalization of $C$. We prove that this locus is nonempty of the expected dimension $2$ if and only if a certain Brill-Noether number is non negative, and that, when nonempty, it gives the desired examples. This is part of a work in progress with Andreas Leopold Knutsen.