Floer homology of surgeries on two-bridge knots

A concordance invariant from the Floer homology of +/- 1 surgeries

We discuss a concordance invariant constructed from Heegaard Floer homology "correction terms" and +/- 1 surgeries on knots in the three-sphere.

Monopoles and lens space surgeries

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.

Finite knot surgeries and Heegaard Floer homology

Extensive rewrite. Tables and proofs have been reformatted and/or rewritten for clarity.

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。

Contact Surgeries on the Legendrian Figure-Eight Knot

We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{\rm{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.

Heegaard Floer groups of Dehn surgeries

We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the number of distinct ranks of the Heegaard Floer groups a Dehn surgery can have. These in turn give a new lower bound on the rational Dehn surgery genus of a rational homology 3-sphere. We also provide novel obstructions for a knot to be a potential counterexample to the Cabling Conjecture.

Heegaard Floer homologies for (+1) surgeries on torus knots

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as the genus and the Levine--Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard Floer homologies of (+1) and (-1) surgeries on torus knots. This relation is best seen at the level of $τ$ functions.

Reducible surgeries and Heegaard Floer homology

In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fact that we show additionally for any knot of genus at most two.

通过增广弗洛理论研究接触(+1)-手术的代数过度扭转性

Algebraic overtwistedness of contact $(+1)$-surgeries via augmented Floer theories

该研究通过增广弗洛理论，证明满足特定Thurston-Bennequin数条件的勒让德n-球面的接触(+1)-手术所得流形接触同调消失，还扩展了三维情形下勒让德纽结接触手术的代数过度扭转性结果。

接触手术与横结的 Cosmetic 问题及结补集问题

Cosmetic contact surgeries along transverse knots and the knot complement problem

研究了标准接触3球体中沿横结的Cosmetic接触手术，证明了非平凡手术的排除，并展示了横结由外皮接触同构类型确定的结论。

Heegaard Floer手术公式与美容手术

Heegaard Floer Surgery Formula and Cosmetic Surgeries

本文利用Heegaard Floer同调公式，证明了特定结的美容手术系数必须大于1，并且非整数手术不会与原流形同胚。

Heegaard Floer correction terms of $(+1)$-surgeries along $(2,q)$-cablings

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cable of any knot $K$. This estimate does not depend on the knot type of $K$. If $K$ belongs to a certain class which contains all negative knots, then equality holds. As a corollary, we show that the relationship between $d_1$ and the Heegaard Floer $τ$-invariant is very weak in gen…

SL(2,C) Floer cohomology for surgeries on some knots

We establish a relationship between the sheaf-theoretic SL(2,C) Floer cohomology HP(Y), as defined by Abouzaid and Manolescu, for Y a surgery on a small knot in S^3, and the SL(2,C) Casson invariant, as defined by Curtis. We determine a similar formula involving the framed sheaf-theoretic SL(2,C) Floer cohomology, HP_#(Y). We use these relationships to compute HP and HP_# for most surgeries on two-bridge knots. We also compute HP for surgeries on two non-small knots, the granny and square knots.

Floer homology and covering spaces

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.

A cube of resolutions for knot Floer homology

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

共享四种手术的纽结

Knots that share four surgeries

本文通过构造共享四个不同斜率 Dehn 手术的纽结对，以及具有接触同胚接触手术的非同位素 Legendrian 纽结对，分别反驳了纽结理论和接触几何中关于手术共享唯一性的两个猜想。

Knot Floer homology of fibred knots and Floer homology of surface diffeomorphisms

We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.

Heegaard Floer homologies of pretzel knots

We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.

Classification of tight contact structures on surgeries on the figure-eight knot

Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.

Dehn surgery, rational open books and knot Floer homology

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgerie…

Algebras with matchings and knot Floer homology

Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the holomorphic theory to bordered Heegaard diagrams for partial knot projections, and establish a pairing result for gluing such diagrams, in the spirit of the pairing theorem of bordered Floer homology. After making some model calculations, we obtain an identification of a variant…

Legendrian knots, transverse knots and combinatorial Floer homology

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

纽结上的手术与紧切触结构

Surgeries on knots and tight contact structures

本文证明对任意纽结和正有理数进行负手术得到的流形总具有紧切触结构，其紧性由非零的Heegaard Floer切触不变量检测。

Transverse Knots Distinguished by Knot Floer Homology

We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.

L-空间结上手术的带标架瞬子同调

Framed instanton homology of surgeries on L-space knots

本文计算了具有瞬子L-空间手术的纽结在所有整数手术上的带标架瞬子Floer同调，并证明若纽结同时具有Heegaard Floer和瞬子Floer的L-空间手术，则两种理论在所有整数手术上一致。

An overview of knot Floer homology

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of k…

All exceptional surgeries on alternating knots are integral surgeries

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

Double knot surgeries to S^4 and S^2 x S^2

It is known that S^4 is a union of two fishtails, and S^2 x S^2 is a union of two cusps (glued along their boundaries). Here we prove that, for any choice of knots K,L in S^3, performing knot surgery operations to S^4 and S^2 x S^2, along both of the fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds.

$L(n,1)$ 型透镜空间之间的手术与 Heegaard Floer $d$-不变量

Surgeries between lens spaces of type $L(n,1)$ and the Heegaard Floer $d$-invariant

本文为 $L$-空间纽结建立了 $d$-不变量手术公式，借此分类了 $L(n,1)$ 型透镜空间间的距离一手术，并应用于环面链环 $T(2,n)$ 间的带手术，证明了其手性美容带化仅当 $n=1,5,9,10$ 时可能发生。

Symmetries and adjunction inequalities for knot Floer homology

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [1] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies. Furthermore, using symmetries we prove that the knot Floer homology of a fiber distinguishes $\stwo\times\sone$ from other…

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…

Heegaard Floer homology and cosmetic surgeries in $S^3$

If a knot $K$ in $S^3$ admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either $\pm 2$ or $\pm 1/q$ for some value of $q$ that is explicitly determined by the knot Floer homology of $K$. Moreover, in the former case the genus of $K$ must be two, and in the latter case there is bound relating $q$ to the genus and the Heegaard Floer thickness of $K$. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to two. We also show that the conjecture holds for any knot…

A note on knot Floer homology of links

Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some constructions related to foliations. We also interpret a theorem of Kauffman in the world of knot Floer homology, hence we can compute the top filtration term of the knot Floer homology for alternative links.

An unoriented skein exact triangle for knot Floer homology

Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating knot, the total rank of its knot Floer homology is equal to the determinant of the knot.

Transverse braids and combinatorial knot Floer homology

We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links that naturally appears in braid diagrams. In this paper, we improve the previous approach in order to compute the transverse invariant. We define a new combinatorial complex that computes knot Floer homology and identify the BRAID invariant of transverse knots and links in the ho…

The classification of Toroidal Dehn surgeries on Montesinos knots

Exceptional Dehn surgeries have been classified for 2-bridge knots and Montesinos knots of length at least 4. In this paper we classify all toroidal Dehn surgeries on Montesinos knots of length 3.

Splicing knot complements and bordered Floer homology

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Heegaard Floer homology of surgeries on two-bridge links

We give an $O(p^{2})$ time algorithm to compute the generalized Heegaard Floer complexes $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s for a two-bridge link $\overrightarrow{L}=b(p,q)$ by using nice diagrams. Using the link surgery formula of Manolescu-Ozsváth, we also show that ${\bf HF}^{-}$ and their $d$-invariants of all integer surgeries on two-bridge links are determined by $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s. We obtain a polynomial time algorithm to compute ${\bf HF}^{-}$ of all the surgeries on two-bridge links, with $\mathbb{Z}/2\mathbb{Z}$ coefficients. In addition, we calculat…

Monopole Floer homology and Legendrian knots

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

Cable knots do not admit cosmetic surgeries

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

On knot Floer homology and cabling II

We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and lens space (or, more generally, L-spac…

A combinatorial description of knot Floer homology

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

On hyperbolic knots realizing the maximal distance between toroidal surgeries

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5.

Knots with unknotting number one and Heegaard Floer homology

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one.

Hyperbolic knots with three toroidal Dehn surgeries

It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed three manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum number. Interestingly, those surgeries correspond to consecutive integers.

Knots in handlebodies with nontrivial handlebody surgeries

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

Non left-orderable surgeries on negatively twisted torus knots

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.

Hyperbolic, L-space knots and exceptional Dehn surgeries

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.

Connected sums of knots do not admit purely cosmetic surgeries

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

Alternating odd pretzel knots and chirally cosmetic surgeries

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2,5) and (2,7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.

Heegaard Floer同调与链环的整手术

Heegaard Floer homology and integer surgeries on links

该研究提出链环的超盒完全系统概念，用以描述链环整手术的Heegaard Floer同调，还给出用四色配边链环表示闭四维流形的方法并结合该系统描述其Ozsvath-Szabo混合不变量，同时说明网格图可生成对应链环的超盒完全系统。

Purely cosmetic surgeries and pretzel knots

We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a pretzel knot provide different oriented three-manifolds.

Chirally Cosmetic Surgeries on Kinoshita-Terasaka and Conway knot families

In this note, we prove that a nontrivial Kinoshita-Terasaka or Conway knot does not admit chirally cosmetic surgeries, by calculating the finite type invariant of order 3.