Splicing integer framed knot complements and bordered Heegaard Floer homology

TL;DR

This paper investigates when gluing two integer-framed knot complements results in an L-space, extending previous results to arbitrary integer framings and characterizing the conditions under which L-spaces are produced.

Contribution

It generalizes the known criteria for L-space formation from 0-framed to arbitrary integer-framed knot complements using bordered Heegaard Floer invariants.

Findings

Splicing two nontrivial knot complements yields an L-space only if both are L-space knots.

The framings must be within a specific range for the resulting manifold to be an L-space.

The analysis relies on detailed bordered Heegaard Floer invariant calculations.

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.