  The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)]
provide the most general known microscopic construction of topological phases
of matter. Based heavily on the mathematical structure of category theory, many
of the special properties of these models are not obvious. In the current
paper, we present a geometrical space-time picture of the partition function of
the Levin-Wen models which can be described as doubles (two copies with
opposite chiralities) of underlying Anyon theories. Our space-time picture
describes the partition function as a knot invariant of a complicated link,
where both the lattice variables of the microscopic Levin-Wen model and the
terms of the hamiltonian are represented as labeled strings of this link. This
complicated link, previously studied in the mathematical literature, and known
as Chain-Mail, can be related directly to known topological invariants of
3-manifolds such as the so called Turaev-Viro invariant and the
Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle
excitations of the Levin-Wen models and we see how they can be understood by
adding additional strings to the Chain-Mail link representing quasi-particle
world-lines. Our construction gives particularly important new insight into how
a doubled theory arises from these microscopic models.
