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We explain how to use bordered algebras to compute a version of link Floer homology. As a corollary, we also give a fast computation of the Thuston polytope for links in the three-sphere.
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Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called bordered knot Floer homology, which has an explicit algebraic and combinatorial construction. In the present paper, we extend the holomorphic theory to bordered Heegaard diagrams for partial knot projections, and establish a pairing result for gluing such diagrams, in the spirit of the pairing theorem of bordered Floer homology. After making some model calculations, we obtain an identification of a variant of knot Floer homology with its algebraically defined relative. These results give a fast algorithm for computing knot Floer homology.
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish.
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 $K_isubset Y_i$, $i=1,2$, 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 $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two differe
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