  We generalize the string-net construction to multiple flavors of strings,
each of which is labeled by the elements of an abelian group $G_i$. The same
flavor of strings can branch while different flavors of strings can cross one
another and thus they form intersecting string-nets. We systematically
construct the exactly soluble lattice Hamiltonians and the ground state wave
functions for the intersecting string-net condensed phases. We analyze the
braiding statistics of the low energy quasiparticle excitations and find that
our model can realize all the topological phases as the string-net model with
group $G=\prod_i G_i$. In this respect, our construction suggests several ways
of building lattice models which realize topological order $G$. They correspond
to intersecting string-net models with various choices of flavors of strings
associated with different decomposition of $G$. In fact, our construction
concretely demonstrates the K\text{\"u}nneth formula by constructing various
lattice models with the same topological order. As an example, we construct the
$G=\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}_2 $ string-net model which
realizes a non-abelian topological phase by properly intersecting three copies
of toric codes.
