Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras

We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C^*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C^*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C^*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.