Floer homology and splicing knot complements

الأصل · EN

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold Y(K₁,K₂) obtained by splicing the complements of the knots Kᵢ⊂ Yᵢ, i=1,2, in terms of the knot Floer homology of K₁ and K₂. We also present a few applications. If hₙⁱ denotes the rank of the Heegaard Floer group HFK for the knot obtained by n-surgery over Kᵢ we show that the rank of HF(Y(K₁,K₂)) is bounded below by |(h∞¹-h₁¹)(h∞²-h₁²)- (h₀¹-h₁¹)(h₀²-h₁²)|. We also show that if splicing the complement of a knot K⊂ Y with the trefoil complements gives a homology sphere L-space then K is trivial and Y is a homology sphere L-space.