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

On computation of morphism spaces and a direct limit of the bordered Floer homology of knot complements