Abstract. In [16], Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.

Using a combinatorial approach described in a recent paper of Manolescu, Ozsváth, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the τ invariant for knots through 11 crossings. We review the basic construction of [3], giving two examples that can be worked out by hand, and explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, closely examining the Kinoshita-Teraska knot KT 2,1 and its Conway mutant.

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We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over $\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page of this spectral sequence is an algorithmically computable chain complex expressed in terms of spanning trees, and we show that there are no higher differentials. This gives the first combinatorial spanning tree model for knot Floer homology.Comment: 58 pages, 18 figures. Published version, with updated reference

Abstract. We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in [1]. IntroductionFloer-theoretic invariants of contact manifolds have been responsible for many important results in low-dimensional topology. Notable examples include the invariants of closed contact 3-manifolds defined by Kronheimer and Mrowka [10] and by Ozsváth and Szabó [18] in monopole and Heegaard Floer homology, respectively. Also important is the work in [8], where Honda, Kazez, and Matić extend Ozsváth and Szabó's construction, using sutured Heegaard Floer homology to define an invariant of sutured contact manifolds, which are triples of the form (M, Γ, ξ) where (M, ξ) is a contact 3-manifold with convex boundary and Γ ⊂ ∂M is a multicurve dividing the characteristic foliation of ξ on ∂M . Recently, we defined an analogous invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured monopole Floer homology theory [1].The goal of this paper is to define an invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured instanton Floer homology (SHI). To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. Like the Heegaard Floer invariants but in contrast with the monopole invariants, our instanton Floer contact invariant is defined using the full relative Giroux correspondence. Its construction is inspired by a reformulation of the monopole Floer invariant in [1] which was used there to prove that the monopole invariant is well-defined.A unique feature of the instanton Floer viewpoint is the central role played by the fundamental group. Along these lines, we conjecture a means by which our contact invariant in SHI might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings, a relationship which has been largely unexplored to this point. Below, we sketch the construction of our contact invariant, describe some of its most important properties, state some conjectures, and discuss plans for future work which include using the constructions in this paper to define invariants of bordered manifolds in the instanton Floer setting.1.1. A contact invariant in SHI. Suppose (M, Γ) is a balanced sutured manifold. Roughly speaking, a closure of (M, Γ) is formed by gluing on some auxiliary piece a...

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, φ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T 2 -bundle over S 1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.Along the way, we compute the Heegaard Floer homology of the torus bundles M T ,φ with b 1 ( M T ,φ ) = 1. Torus bundles are of interest in other Floer theories as well. For instance, when φ is the identity, Hutchings and Sullivan show in [15] that the embedded contact homology of M T ,φ ∼ = T 3 agrees with its Heegaard and Seiberg-Witten Floer homologies up to shifts in grading, evidence for the conjecture that these three Floer theories are equivalent despite their very different constructions. Eli Lebow has since computed the embedded contact homology for almost all T 2 -bundles over S 1 (see [20]). In § 6, we compare our results with his and verify the following.Theorem 1.1. Embedded contact homology and Heegaard Floer homology are isomorphic as relatively graded Z-modules for any T 2 -bundle over S 1 with pseudo-Anosov monodromy.Surface bundles over S 1 have also been studied in the context of Heegaard Floer homology by Jabuka and Mark in [16,17,18], and by Roberts in [45]. As we shall see, our computations may be used to extend some of their results on the Heegaard Floer homology in nontorsion Spin c structures of Σ g -bundles over S 1 , where Σ g is a closed surface of genus g > 1.Below, we discuss some topological and geometric applications of our computations. L-spaces and quasi-alternating linksAn important step in our computations of HF + (M T ,φ ) and HF + ( M T ,φ ) is the identification of all L-spaces among the manifolds M T ,φ .Definition 1.2. An L-space is a rational homology 3-sphere whose Heegaard Floer homology is as simple as possible; namely, HF (Y, s) ∼ = Z, or equivalently,There is no purely topological characterization of L-spaces, though L-spaces do come with some rigid geometric restrictions. Most notably, Ozsváth and Szabó prove that if Y is an L-space, then Y contains no co-orientable taut foliation (see [34]). In [2], we use this result to produce an infinite family of hyperbolic 3-manifolds with no co-orientable taut foliations, extending the first known set of such 3-manifolds, which was discovered by Roberts, Shareshian, and Stein among fillings of punctured torus bundles (see [48]).L-spaces are commonly found among branched covers of S 3 . Ozsváth and Szabó prove, for instance, that if K ⊂ S 3 is an alternating link then Σ(K), the double cover of S 3 branched along K, is an L-space (see [40]). In the same pap...

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić's contact invariant in sutured Heegaard Floer homology (SFH). In the process of defining our invariant, we construct maps on SHM associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in SFH. We use these maps to establish a bypass exact triangle in SHM analogous to Honda's in SFH. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939-978] to define a contact invariant in the instanton Floer setting.

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots -namely, the range of slopes on which admissible transverse surgery preserves tightness -and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.