In this talk we consider the (reduced) Donaldson-Thomas theory of the product of a K3 surface and an elliptic curve. In rank 1, these invariants can be viewed as enumerating algebraic curves and are known for fiber classes over the elliptic curve (work of Pandharipande and Thomas), and for classes primitive over the K3 (work of Pixton, Shen and myself). I will explain how to extend these results to arbitrary curve classes. If time permits, I will also give an outlook on the higher rank case (work in progress).

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