  We adapt the bialgebra and Hopf relations to expose internal structure in the
ground state of a Hamiltonian with $Z_2$ topological order. Its tensor network
description allows for exact contraction through simple diagrammatic rewrite
rules. The contraction property does not depend on specifics such as geometry,
but rather originates from the non-trivial algebraic properties of the
constituent tensors. We then generalise the resulting tensor network from a
spin-1/2 lattice to a class of exactly contractible states on spin-S degrees of
freedom, yielding the most efficient tensor network description of finite
Abelian lattice gauge theories. We gain a new perspective on these states as
examples of two-dimensional quantum states with algebraically contractible
tensor network representations. The introduction of local perturbations to the
network is shown to reduce the von Neumann entropy of string-like regions,
creating an unentangled sub-system within the bulk in a certain limit. We also
show how perturbations induce finite-range correlations in this system. This
class of tensor networks is readily translated onto any lattice, and we
differentiate between the physical consequences of bipartite and non-bipartite
lattices on the properties of the corresponding quantum states. We explicitly
show this on the hexagonal, square, kagome and triangular lattices.
