Splicing knot complements and bordered Floer homology

TL;DR

This paper proves that splicing two nontrivial knot complements in integer homology sphere L-spaces results in a manifold with more complex Heegaard Floer homology, showing such splicing cannot produce L-spaces.

Contribution

It demonstrates that splicing nontrivial knot complements in integer homology sphere L-spaces increases Floer homology rank, using bordered Floer homology techniques.

Findings

Splicing two nontrivial knot complements yields a Heegaard Floer homology rank greater than one.

Splicing nontrivial knots in the 3-sphere does not produce L-spaces.

Bordered Floer homology is used to establish these results.

Abstract

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.