Floer homology and splicing knot complements

TL;DR

This paper derives a formula for the Heegaard Floer homology of 3-manifolds formed by splicing knot complements, linking it to the knot Floer homology of the individual knots, with applications to bounds and characterizations of L-spaces.

Contribution

It provides a new formula for Heegaard Floer homology of spliced manifolds and explores implications for L-spaces and knot triviality.

Findings

The rank of the Floer homology is bounded below by a specific algebraic expression.

Splicing with trefoil complements yields conditions for knots to be trivial and manifolds to be L-spaces.

The paper offers applications to bounds on Floer homology and characterizations of L-space knots.

Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $\widehat{\mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $\widehat{\mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$\big|(h_\infty^1-h_1^1)(h_\infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)\big|.$$ We also show that if splicing the complement of a knot $K\subset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.