A refinement of sutured Floer homology

Abstract

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.

A Survey of Heegaard Floer Homology

“…He obtains various nice gluing results in this context. The second is a generalization of the minus version of Heegaard Floer homology to sutured manifolds, due to Alishahi and Eftekhary [1]. The chain complex they define is over an algebra depending on the sutures, and is well-defined up to chain homotopy equivalence.…”

Knot Floer homology and Khovanov–Rozansky homology for singular links

“…Since the α circle contains O i , it gives a coefficient of U i , and similarly, the β circle gives a coefficient of U j . It was shown in [1] that the moduli spaces can be oriented such that the α and β degenerations come with opposite signs, so choosing such a sign convention, this gives d 2 = ±(U i − U j )I Moreover, the additional differentials are also subject to the basepoint filtration, so we get d 2 0 = ±(U i − U j )I This X basepoint lies inside a minimal bigon, and this bigon now contributes to the differential with a coefficient of ±1. The local contribution therefore must be the the complex in Figure 7, so U i and U j must come with opposite sign.…”

Peculiar modules for 4‐ended tangles

“…Let us start by recalling the basic definitions from [41,Sections 1,4], adapted to 4-ended tangles. tangle end and following the orientation of the fixed circle S 1 , we number the tangle ends and label the arcs S 1 im(T ) by a, b, c and d, in that order.…”

“…In fact, the definition of our generalised peculiar modules is primarily inspired by the invariants from , see Remark . In , Eftekhary and Alishahi define a Heegaard Floer theory for tangles using a suitable generalisation of sutured Floer homology . They study cobordism maps between their tangle invariants, but they do not discuss glueing properties.…”