  This is the third in a series of papers studying the vertex-algebraic
structure of principal subspaces of twisted modules for lattice vertex operator
algebras. We focus primarily on lattices $L$ whose Gram matrix contains only
non-negative entries. We develop further ideas originally presented by
Calinescu, Lepowsky, and Milas to find presentations (generators and relations)
of the principal subspace of a certain natural twisted module for the vertex
operator algebra $V_L$. We then use these presentations to construct exact
sequences involving this principal subspace, which give a set of recursions
satisfied by the multigraded dimension of the principal subspace and allow us
to find the multigraded dimension of the principal subspace.
