Splicing integer framed knot complements and bordered Heegaard Floer homology

Abstract

We consider the following question: when is the manifold obtained by gluing together two knot complements an L-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an L-space. We extend this result to allow for arbitrary integer framings. We find that splicing two integer framed nontrivial knot complements only produces an L-space if both knots are L-space knots and the framings lie in an appropriate range. The proof involves a careful analysis of the bordered Heegaard Floer invariants of each knot complement.