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We study the reduced descendent Gromov-Witten theory of K3 surfaces inprimitive curve classes. We present a conjectural closed formula for thestationary theory, which generalizes the Bryan-Leung formula. We also prove anew recursion that allows to remove descendent insertions of $1$ in manyinstances. Together this yields an efficient way to compute a large class ofinvariants (modulo the conjecture on the stationary part). As a corollary weconjecture a surprising polynomial structure which underlies the Gromov-Witteninvariants of the K3 surface.

On the descendent Gromov-Witten theory of a K3 surface