Zhongtao Wu
In this paper, we write down a special Heegaard diagram for a given product three manifold $Σ_g\times S^1$. We use the diagram to compute its perturbed Heegaard Floer homology.
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We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.
We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In particular, we show that certain polynomials are never the Alexander polynomials of knots which admit homology three-sphere Seifert fibered surgeries. The knot Floer homology restrictions, on the other hand, apply also in cases where the Alexander polynomial gives no information, s…
Modifying the method of [21], we compute the perturbed $HF^+$ for some special classes of fibered three manifolds in the second highest spin$^c$-structures $S_{g-2}$. The special classes considered in this paper include the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves.
For a Liouville domain $W$ whose boundary admits a periodic Reeb flow, we can consider the connected component $[τ] \in π_0(\text{Symp}^c(\widehat W))$ of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component $[τ]$ and give a uniform lower bound of the growth using wrapped Floer homology. We also show that $[τ]$ has infinite order in $π_0(\text{Symp}^c(\widehat W))$ if there is an admissible Lagrangian $L$ in $W$ whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor …
We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3--sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3--manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also intr…
We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also these groups for other three-manifolds, including the product of a circle with a genus two surface.
Grid Diagrams of Fibered Knots
研究纤维化结的特殊网格图存在性，开发高效方法判断网格图是否满足唯一网格状态实现Alexander函数上界条件，并发现5385个纤维化素结的合适网格图。
We classify tight contact structures on the small Seifert fibered 3--manifold M(-1; r_1, r_2, r_3) with r_i in (0,1) and r_1, r_2 \geq 1/2. The result is obtained by combining convex surface theory with computations of contact Ozsvath--Szabo invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.
If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$ fixed points. This generalizes earlier work of Baldwin--Hu--Sivek and Ni. We also clarify a misleading formula in Cotton-Clay's computation of the symplectic Floer homology of mapping classes of surfaces.
Floer homology and right-veering monodromy
本文证明纽结Floer复形可检测纤维化纽结的单值是否右转向，从而给出紧接触结构的纯Floer理论刻画，并应用于Dehn手术与紧叶状结构。
For an open book decomposition $(S,ϕ)$, the fractional Dehn twist coefficients are rational numbers measuring the amount that the monodromy $ϕ$ twists the surface $S$ near each boundary component. In general, the twist coefficients do not behave nicely under the operation of capping off a boundary component. The goal of this paper is to use Heegaard Floer homology to constrain the behavior of the fractional Dehn twist coefficients after capping off. We also use our results about fractional Dehn twists to study the Floer homology of cyclic branched covers over fibered two-component links.
This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the author has constructed on general symplectic manifolds [Oh4,5]. The existence theorem is also the crux of the author's recent proof of an optimal energy-capacity inequality given in [Oh5]. In this paper, we prove a more general existence result than needed in that we study Floer'…
Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot $K$. As an application, we express the Heegaard Floer homology of rational surgeries on $Y$ along a null-homologous knot $K$ in terms of the filtered homotopy type of the knot invariant for $K$. This has applications to Dehn surgery problems for knots in $S^3$. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homolo…
We develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves whose domains are either strips or more general Riemann surfaces with strip-like ends. These moduli spaces can be injectively immersed into the Hilbert manifold $\mathcal{P} $ by taking the restriction to non-Lagrangian boundary. Some subsets $ \mathcal{M}^T(U) $ and $ \mathcal{M}^{\infty} (U) $ of the aforementioned moduli spaces of perturbed holomorphic stri…
We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These "relative adjunction inequalities" improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every n…
Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence
本文研究预量子化丛的分裂 Rabinowitz Floer 同调，在 Liouville 填充与辛化两种情形下构造联系量子同调的 Gysin 型正合序列，并用于可逆性、同痕、可序化及平移点问题。
Towards a symplectic Khovanov homology for links in fibered $3$-manifolds
本文针对纤维化闭3-流形中的横向链环定义辛Khovanov型同调，并猜想相关曲面范畴的组合dga描述，该dga是带边Heegaard Floer同调中链代数的高维类似物。
In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita-Terasaka knots and their ``Conway mutants''. These results show that $\HFKa$ contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that $\HFKa$ detects mutation. We also calculate $\HFKa$ for certain pretzel knots, and knots with small crossing number ($n\leq 9$). Our calculatio…
Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, …
Legendrian contact instanton cohomology and its spectral invariants on the one-jet bundle
本文发展了Legendrian链相关哈密顿扰动切触作用泛函的Floer型椭圆Morse理论，构造了Legendrian切触瞬子上同调及其谱不变量，涵盖了此前余切丛上的相关Lagrangian相交理论成果。
We define the $S^1$-equivariant Rabinowitz-Floer homology of a bounding contact hypersurface $Σ$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $Σ$ is displaceable. In the appendix we describe an approach to transversality for Floer homologies for which the moduli space $\widehat M_J$ of all gradient flow lines is compact for some almost complex structure $J$. This approach uses a large set of perturbations, namely vector fields on the loop space, and selects from the possibly non-compact perturbed moduli spaces a part near $\widehat M_J$ that turns out t…
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exact triangle for fibered Dehn twists due to Wehrheim and Woodward. As applications we obtain a spectral sequences from Khovanov homology to symplectic Khovanov homology. Also when a 3-manifold $M$ is given by gluing two handlebodies by a surface diffeomorphism $ϕ$, we obtain a spe…
In a previous paper, the author introduced a Floer-theoretic torsion invariant I_F, which roughly takes the form of a product of a power series counting perturbed pseudo-holomorphic tori, and the Reidemeister torsion of the symplectic Floer complex. We pointed out the formal resemblance of I_F with a generating function of genus 1 Gromov invariant; furthermore, for heuristic reasons one also expects a relation with the 1-loop generating function in the A-model side of mirror symmetry, which counts genus 1 holomorphic curves. The present article makes this expected relation precise in the simpl…
In this paper we will prove that for a compact, symplectic manifold $(M, ω)$ and for $ω$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to redefine the Floer symplectic homology. In particular, in subsequent papers, we will prove the Arnold conjecture in both degenerate and non-degenerate cases with integer coefficients for general, symplectic manifolds.
Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.
We introduce in this paper a field theory on symplectic manifolds that are fibered over a real surface with interior marked points and cylindrical ends. We assign to each such object a morphism between certain tensor products of quantum and Floer homologies that are canonically attached to the fibration. We prove a composition theorem in the spirit of QFT, and show that this field theory applies naturally to the problem of minimising geodesics in Hofer's geometry. This work can be considered as a natural framework that incorporates both the Piunikhin-Salamon-Schwarz morphisms and the Seidel is…
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