  In the bordered Floer theory, gluing thickened torus of positive meridional
Dehn twist to the boundary of a knot complement result in the knot complement
of increased framing. For a fixed knot K, we construct a direct system of
positively framed knot complements and study the direct limit. We also study
the morphism space between two type-DD modules, and derive type-DA morphisms
from DD morphisms to derive the direct system maps. In addition, we introduce a
direct limit invariant from the direct system which can detect non-unstable
chains in the type-D module of a knot complement, if the type-D modules of the
direct system are obtained by algorithm of Lipshitz, Ozsvath and Thurston.
