  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 k-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.
