Longitude Floer homology and the Whitehead double

Legendrian 非简单 Trefoil 的 Whitehead 双曲线

Legendrian Non-simple Whitehead Doubles Of The Trefoil

研究显示 Trefoil 的 Whitehead 双曲线在标准接触 3 球面中为 Legendrian 非简单，利用结 Floer 同调和手术三角形证明其非简单性。

奇异\(\mathbb{R}^4\)、RBG 链环与端弗洛尔同调

Exotic $\mathbb{R}^4$'s, RBG Links, and End Floer Homology

研究通过将微分同胚卡森柄附于微分同胚圆盘补集构造非微分同胚奇异\(\mathbb{R}^4\)，利用 RBG 链环构造找特殊切片纽结，借助端弗洛尔同调区分奇异\(\mathbb{R}^4\)。

Knot doubling operators and bordered Heegaard Floer homology

We use bordered Heegaard Floer homology to compute the tau invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of Whitehead doubling. We show that tau of the resulting knot depends only on the two twisting parameters and the values of tau for the two companion knots. We also include some notes on bordered Heegaard Floer homology that may serve as a useful introduction to the subject.

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。

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called Longitude Floer Homology in a way that enables us to byp…

L-空间卫星算子与纽结 Floer 同调

L-space satellite operators and knot Floer homology

本文针对对应二分支 L-空间链环的卫星算子，利用 Heegaard Floer 手术公式证明其卫星纽结 Floer 复形可由原纽结复形计算，并提供 Python 实现。

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…

Bordered Floer homology and existence of incompressible tori in homology spheres

Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of the bordered Floer complex of $Y(K)$ in terms of the knot Floer complex associated with $K$. As a corollary, we show that if a homology sphere has the same Heegaard Floer homology as $S^3$ it does not contain any incompressible tori. Consequently, if $Y$ is an irreducible homology sphere $L$-space then $Y$ is either $S^3$, or the Poicaré sphere $Σ(2,3,5)$, or…

On the twisted Floer homology of mapping tori of periodic diffeomorphisms

Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particula…

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…

The Ozsv{á}th-Szab{ó} and Rasmussen concordance invariants are not equal

In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{á}th-Szab{ó} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.

结手术公式用于瞬子Floer同调II：应用

Knot surgery formulae for instanton Floer homology II: applications

本文通过整数手术公式研究了3流形的瞬子同调，揭示了几乎L空间结的纤维化和强准正性质，并确定了特定类型的结结构。

Khovanov polynomials for satellites and asymptotic adjoint polynomials

We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companio…

小Seifert L-空间及更多类别的三维辛化问题

The three-dimensional symplectization question for small Seifert L-spaces and further classes

本文通过构造Liouville配边，解决了小Seifert L-空间等若干三维流形的辛化问题，证明了强可填充性等刚性性质。

加倍圆盘与卫星曲面

Doubled Disks and Satellite Surfaces

本文研究切片圆盘的正怀特黑德加倍与光滑同位素的关系，证明多种同调阻碍在卫星操作下保持，并据此构造出四维球中大量奇异圆盘及首个两两奇异切片圆盘的无限族。

一个4维有理亏格界

A 4-dimensional rational genus bound

本文引入纽结的有理切片亏格，用Heegaard Floer τ不变量给出下界，并应用于Floer简单纽结及产生切片亏格无界序列等。