Heegaard Floer homology, degree-one maps and splicing knot complements

TL;DR

This paper investigates how the Heegaard Floer homology rank behaves under the splicing of knot complements in homology spheres, establishing a lower bound related to the original manifold.

Contribution

It proves that the rank of the Heegaard Floer homology of the spliced manifold is at least as large as that of the original homology sphere.

Findings

Rank of -hat increases or stays the same after splicing.

Provides a lower bound for the Heegaard Floer homology rank post-splicing.

Enhances understanding of knot complement operations in 3-manifold topology.

Abstract

Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.