Heegaard Floer homologies of pretzel knots

Heegaard Floer homology of Some Mazur Type Manifolds

We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsvath--Szabo invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard--Floer homologies. We also show that the rank of the Heegaard--Floer homology can get arbitrary large values in this family by using its relation with the Casson invariant. For comparison, we list the ranks of Floer homologies of all the examples of Briekorn spheres that are known to bound contractible manifold…

Lattice and Heegaard-Floer homologies of algebraic links

We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a certain variant of the Orlik-Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of al…

Heegaard Splittings and Seiberg-Witten monopoles

This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which exhibits characteristics of both theories.

Pretzel knots with L-space surgeries

A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.

Algebraic torsion via Heegaard Floer homology

We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.

Généralisation de l'homologie d'Heegaard-Floer aux entrelacs singuliers & Raffinement de l'homologie de Khovanov aux entrelacs restreints

A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i.e. links with a finite number of rigid trans…

分支覆盖的Heegaard Floer同调的秩不等式

Rank inequalities for the Heegaard Floer homology of branched covers

本文研究沿零同调链环分支的二重覆盖，建立了Heegaard Floer同调维数的不等式，并探讨其与L-空间猜想的关系及拓扑应用。

The Thurston polytope for four-stranded pretzel links

In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly construct norm-realizing examples of such surfaces.

Genus two mutant knots with the same dimension in knot Floer and Khovanov homologies

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.

Combinatorial cobordism maps in hat Heegaard Floer theory

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.

Complexity, Heegaard diagrams and generalized Dunwoody manifolds

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and…

On the slice-ribbon conjecture for pretzel knots

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard-Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson-Gordon invariants…

Heegaard Floer homology of certain mapping tori

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that …

Heegaard Floer homology and splicing homology spheres

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and only if the Heegaard Floer homologies are isomorphic.

Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology

Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.

从 Heegaard-Floer 视角看完整性

Holonomicity from a Heegaard-Floer Perspective

本文构造 $S^r$ 着色纽结 Floer 同调并建立范畴化递推，从而由欧拉示性数推出着色 Alexander 多项式序列的 $q$ 完整性。

纽结Floer同调中的连通和与有向系统

Connected sums and directed systems in knot Floer homologies

本文证明了instanton knot Floer homology的基本性质，主要贡献包括推导其连通和公式、有向skein正合三角形，以及提出适用于所有Floer理论的自由有限生成链复形新模型。

Heegaard Floer 同调与手性装饰手术

Heegaard Floer homology and chirally cosmetic surgeries

本文利用 Heegaard Floer 同调中的浸入曲线表述和手术公式，为纽结的手性装饰手术建立新阻碍，并据此完全分类奇交错 pretzel 纽结、排除大量 Whitehead 双结及 L-space 纽结的相关手术。

Decomposing sutured monopole and Instanton Floer homologies

In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for …

Whitney polygons, symbol homology and cobordism maps

We define a new homology theory we call symbol homology by using decorated moduli spaces of Whitney polygons. By decorating different types of moduli spaces we obtain different flavors of this homology theory together with morphisms between them. Each of these flavors encodes the properties of a different type of Heegaard Floer homology. The morphisms between the symbol homologies enable us to push properties from one Floer theory to a different one. Furthermore, we obtain a new presentation of Heegaard Floer theory in which maps correspond to multiplication from the right with suitable elemen…

On the tau invariants in instanton and monopole Floer theories

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying $τ_{\mathrm{G}}$, defined by the second author via the minus flavors $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KHM}}^-$ of the knot homologies, with $τ_{\mathrm{G}}^{\sharp}$, defined by Baldwin and Sivek via cobordism maps of the $3$-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KH…

A note on quasi-alternating Montesinos links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we…

On a Heegaard Floer theory for tangles

The purpose of this thesis is to define a "local" version of Ozsváth and Szabó's Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\operatorname{\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\operatorname{\widehat{HFT}}$ and its decategorified tangle invariant $\nabla_T^s$, we prove a glueing theorem in terms of Zarev's bordered sutured Floer homology, which endows $\operatorname{\widehat{HFT}}$ with an additional glueing structure. For 4-ended tangles, we repackage this glueing …

Traceless Character Varieties, the Link Surgeries Spectral Sequence, and Khovanov Homology

In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…

The Knot Floer Cube of Resolutions and the Composition Product

We examine the relationship between the (untwisted) knot Floer cube of resolutions and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT homologies of various subdiagrams. Jaeger's composition product formula shows that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of $K$.

Link homology theories and ribbon concordances

It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.

Ribbon homology cobordisms

We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on the behavior of knot Floer homology under ribbon concordances. Finally, we provide topological applications, including to Dehn surgery problems.

强可逆纽结及其商纽结的Khovanov同调

Khovanov homology of strongly invertible knots and their quotients

本文通过构造联系强可逆纽结与其商纽结的谱序列，重新证明Khovanov同调可区分某些切片圆盘，并给出分支二重覆盖的Heegaard Floer同调类比结果。

Knot lattice homology in L-spaces

We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)$ is isomorphic to the Heegaard Floer homology HF^-(Y_G) of the corresponding rational homology sphere Y_G.

Peculiar modules for 4-ended tangles

With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify peculiar modules in terms of immersed curves on the 4-punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson, we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinte…

Variants of equivariant Seiberg-Witten Floer homology

For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduce…

The cosmetic crossing conjecture for split links

Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the …

Framed instanton homology and concordance

We define two concordance invariants of knots using framed instanton homology. These invariants $ν^\sharp$ and $τ^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $τ$ invariant in Heegaard Floer homology. We use $ν^\sharp$ and $τ^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyper…

框架瞬子同调与结的同调，II

Framed instanton homology and concordance, II

本文研究了框架瞬子同调与结的同调，证明了$ν^\sharp(K)$的性质，并定义了瞬子FloER的类似量$ε^\sharp(K)$，同时探讨了3-流形作为结手术的问题。

On spectral sequences from Khovanov homology

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology…

A class of knots with simple $SU(2)$ representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det(K)$. In fact, the complex underlyin…