We introduce a refinement of the Ozsváth-Szabó complex associated to a balanced sutured manifold (X, τ ) by Juhász [Ju1]. An algebra Aτ is associated to the boundary of a sutured manifold and a filtration of its generators by H 2 (X, ∂X; Z) is defined. For a fixed class s of a Spin c structure over the manifold X, which is obtained from X by filling out the sutures, the Ozsváth-Szabó chain complex CF(X, τ, s) is then defined as a chain complex with coefficients in Aτ and filtered by Spin c (X, τ ). The filtered chain homotopy type of this chain complex is an invariant of (X, τ ) and the Spin c class s ∈ Spin c (X). The construction generalizes the construction of Juhász. It plays the role of CF − (X, s) when X is a closed three-manifold, and the role ofwhen the sutured manifold is obtained from a knot K inside a three-manifold Y . Our invariants generalize both the knot invariants of Ozsváth-Szabó and Rasmussen and the link invariants of Ozsváth and Szabó. We study some of the basic properties of the corresponding Ozsváth-Szabó complex, including the exact triangles, and some form of stabilization.
We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, u(K). In particular, for knots with u(K) ≤ 2, we show that the Lee spectral sequence must collapse at the E 2
We show that bordered Heegaard Floer homology detects homologically essential compressing disks and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Λ so that the tensor product of CFD(Y ) and Λ is Hom-orthogonal to CFD(Y ) if and only if the boundary of Y admits an essential compressing disk. In the process, we sharpen a nonvanishing result of Ni's. We also extend Lipshitz-Ozsváth-Thurston's "factoring" algorithm for computing HF [LOT14] to compute bordered-sutured Floer homology, to make both results on detecting incompressibility practical. In particular, this makes Zarev's tangle invariant manifestly combinatorial. arXiv:1708.05121v2 [math.GT] 30 Aug 2017 4 AKRAM ALISHAHI AND ROBERT LIPSHITZ Finally, note that CFD(Y, φ i ) is algorithmically computable from a Heegaard decomposition of Y [LOT14]. Essentially the same algorithm also computes bordered-sutured Floer homology, though this has not been spelled-out in detail in the literature. The bimodule Λ(Z) is described below explicitly, but involves power series, so it is not immediately obvious if Theorems 1.3 and 1.8 give effective criteria. Nonetheless, the underlying techniques are effective, and we give computationally effective versions of Theorems 1.3 and 1.8 in Section 7.This paper is organized as follows. Section 2 collects some background results on Heegaard Floer homology with Novikov coefficients, sutured Floer homology, bordered Floer homology, and bordered-sutured Floer homology. Section 3 proves the non-vanishing theorems we need for Heegaard Floer homology, Theorems 1.1 and 1.2, as well as a non-vanishing result for sutured Floer homology, Theorem 3.3. Section 4 shows that bordered Floer homology detects essential compressing disks for 3-manifolds with connected boundary, Theorem 1.3. An extension to disconnected boundary is given in Section 5. Section 6 proves that borderedsutured Floer homology detects partly boundary parallel tangles, Theorem 1.8. The long final section of the paper (Section 7) is devoted to showing that these results are effective and, at least plausibly, computationally useful. The main work in that section is extending the "factoring" algorithm for computing HF [LOT14] to compute bordered-sutured Floer homology; this extension seems of independent interest.
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