#### Volume 15, issue 6 (2015)

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Floer homology and splicing knot complements

### Eaman Eftekhary

Algebraic & Geometric Topology 15 (2015) 3155–3213
##### Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y\left({K}_{1},{K}_{2}\right)$ 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 $\stackrel{̂}{HFK}$ for the knot obtained by $n$–surgery over ${K}_{i}$, we show that the rank of $\stackrel{̂}{HF}\left(Y\left({K}_{1},{K}_{2}\right)\right)$ is bounded below by

$|\left({h}_{\infty }^{1}-{h}_{1}^{1}\right)\left({h}_{\infty }^{2}-{h}_{1}^{2}\right)-\left({h}_{0}^{1}-{h}_{1}^{1}\right)\left({h}_{0}^{2}-{h}_{1}^{2}\right)|.$

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.

##### Keywords
Floer homology, splicing, essential torus
Primary: 57M27
Secondary: 57R58