Floer homology and splicing knot complements

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 $$|(h_{\infty }^1-h_1^1)(h_{\infty }^2-h_1^2)-(h_0^1-h_1^1)(h_0^2-h_1^2)|.$$ 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.

Some errors in version 2 of the paper are corrected, and the exposition is slightly improved

References in corpus (7)

• Floer homology and knot complements
• Bordered Heegaard Floer homology: Invariance and pairing
• On combinatorial link Floer homology
• Splicing knot complements and bordered Floer homology
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Cited by in corpus (9)

• Bordered Heegaard Floer homology: Invariance and pairing
• Splicing knot complements and bordered Floer homology
• Bordered Floer homology and existence of incompressible tori in homology spheres
• Instanton L-spaces and splicing
• Correction to the article: Floer homology and splicing knot complements
• Holomorphic polygons and the bordered Heegaard Floer homology of link complements
• Taut foliations, braid positivity, and unknot detection
• Knots which admit a surgery with simple knot Floer homology groups
• Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture