Legendrian Vertical Circles in Small Seifert Spaces

Classical invariants of Legendrian knots in the 3-dimensional torus

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $ξ$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight orie…

某些小 Seifert 纤维空间族上的紧接触结构

Tight contact structures on some families of small Seifert fiber spaces

本文提出一种通过比较等价 Thurston–Bennequin 标架的 Legendrian 表示来证明手术后接触结构仍不同的方法，并结合凸曲面理论分类若干小 Seifert 纤维空间族上的紧接触结构。

辛理性同调球填充的Seifert纤维空间

Symplectic rational homology ball fillings of Seifert fibered spaces

研究了Seifert纤维空间作为辛理性同调球边界条件，证明了特定棱柱空间的填充可能性，并探讨了Brieskorn球体与Stein领域的关系。

Legendrian helix and cable links

Lisa Traynor has described an example of a two-component Legendrian `circular helix link' in the 1-jet space of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to that same link with the order of the two components reversed. We give a complete classification of the Legendrian realisations of this topological link type, as well as all other `cable links' in that 1-jet space.

几何3流形的灵活指数与Seifert空间的Legendrian映射

Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces

本文研究了几何3流形的灵活指数，通过构造Legendrian映射，确定了不同几何结构下流形的灵活指数，证明了某些流形的灵活指数为8/3，并指出Legendrian映射不能是微分同胚。

Rectangular Seifert circles and arcs system

Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R. Cromwell and I. A. Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram $D$ with $c(D)$ crossings and $s(D)$ Seifert circles can be deformed by an ambient isotopy of ${\mathbb R}^2$ into a rectangular diagram w…

Legendrian knots and links classified by classical invariants

It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space of the circle with its standard tight contact structure.

Seifert Manifolds

A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like structures whose fibers are infra- homogeneous spaces; that is, the fibers are flat manifolds, almost flat manifolds, etc. We prove existence, uniqueness, rigidity theorems. Many interesting properties and applications are presented.

Seifert fibered four-manifolds with nonzero Seiberg-Witten invariant

The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surgery, without changing the integral homology, intersection form, and even the Seiberg-Witten invariant. Results concerning classification of Seifert fibered complex surfaces or symplectic 4-manifolds are included. We also show that every smooth circle action on the 4-torus is smoot…

The Braid Index Of Reduced Alternating Links

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid ind…

Seifert圆、交叉数与广义纽结和链环的辫指数

Seifert circles, crossing number and the braid index of generalized knots and links

本文引入广义交叉和广义Reidemeister型移动，研究与交叉类型无关的链环性质，证明了广义纽结和链环总交叉数与辫指数间的不等式，并适用于虚拟奇链环。

Shadows of Wave Fronts and Arnold-Bennequin Type Invariants of Fronts on Surfaces and Orbifolds

A first order Vassiliev invariant of an oriented knot in an $S^1$-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds related to the Bennequin-type invariants of the Legendrian curves studied by F. Aicardi, V. Arnold, M. Polyak, and S. Tabachnikov. Formulas expressing these relations are presented. We also calculate Turaev's shadow for the Legendrian lifting of a wave front. This allows to use in the…

The topology of symplectic circle bundles

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert fibered, has vanishing Thurston norm, or if the total space admits a Lefschetz fibration.

Torsion Invariants of Combed 3-Manifolds with Boundary Pattern and Legendrian Links

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in contact 3-manifolds. Our approach uses a combinatorial encoding of vector fields, based on standard spines. In this paper we extend this encoding from closed manifolds to manifolds with boundary.

Twist families of L-space knots, their genera, and Seifert surgeries

Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $ω$ between $K$ and $c$. We establish the conjecture in case of $|ω| \neq 1$, prove that $\{K_n\}$ contains at most three L-space knots if $ω= 0$, and address the case where $|ω| = 1$ under an additional hypothesis about Seifert surgeries. To that end, we characterize a twisting circle $c$ for which $\{ (K_n, r_n) \}$ contains at least ten S…

Non-isotopic Heegaard splittings of Seifert fibered spaces

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if M has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen's conjecture.

On the complexity of non-orientable Seifert fibre spaces

In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with boundary. Moreover, we compute a potentially sharp upper bound for their complexity in terms of the invariants of the combinatorial description, extending to the non-orientable case results by Fominykh and Wiest for the orientable case with boundary and by Martelli and Petronio for the closed orientable case.

The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of…

A Diagrammatic Approach for Determining the Braid Index of Alternating Links

It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only crossing between two Seifert circles in $D$). If $D$ contains lone crossings, then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not known how $s(D)$ is related to $\textbf{b}(D)$. In this …

Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n. As an application, we determine th…

Reidemeister Torsion of 3-Dimensional Euler Structures with Simple Boundary Tangency and Pseudo-Legendrian Knots

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's H_1(M)-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) pseudo-Legendrian knots (i.e. knots transversal to a given vector field), and hence to Legendria…

Topology of Vortex Reconnection

Knotted vortices such as those produced in water by Kleckner and Irvine tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number $R(K)$ of an oriented knot of link $K$ is the least number of reconnections (oriented re-smoothings) needed to unknot/unlink $K$. Putting this problem into the context of knot cobordism, we show, using Rasmussen's Invariant that the reconnection number of a positive knot is equal to twice the genus of its Seifert spanning surface. In particular an $(a,b)$ torus knot has $R = (a-1)(b-1).$ For an arbitrary unsplittable…

Intersection Pairings on Spaces of Connections and Chern-Simons Theory on Seifert Manifolds

Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. When M is the product of a Riemann surface with a circle the 4-dimensional class does not enter and the path integral takes the form of a Riemann-Roch formula albeit in infinite dimensions. The discussion is generalised to include Wilson lines along the fibre direction in M.

基于Siegel上半空间的Seifert几何的推广

A generalization of Seifert geometry based on the Siegel upper half-space

本文基于Siegel上半空间构造了类似于Seifert几何的几何结构，并给出了具有该几何的流形的体积公式，证明了Siegel-Seifert闭流形的体积等于纤维圆长度乘以底流形的欧拉示性数（带符号）。

三维中的紧接触结构空间

Spaces of tight contact structures in dimension 3

该研究引入微纤维化方法，证明紧接触结构空间的弱同伦型在接触0-、1-、3-配柄附接下不变，还确定了多种接触结构空间及接触同胚群的弱同伦型，部分解决Giroux猜想并证明了Hatcher定理的类似结论。

Legendrian 纽结在海森堡群中的 CR 不变能量

CR-invariant energy of Legendrian knots in the Heisenberg group

本文研究了海森堡群中 Legendrian 纽结的 CR 不变能量，通过正则化 Korányi 距离下的 -2 阶势能积分来定义能量函数，并展示了实数圆作为能量最小化者的特性，同时给出了海森堡群中的 Doyle--Schramm 余弦公式类比。

Localization for Wilson Loops in Chern-Simons Theory

We reconsider Chern-Simons gauge theory on a Seifert manifold M, which is the total space of a nontrivial circle bundle over a Riemann surface, possibly with orbifold points. As shown in previous work with Witten, the path integral technique of non-abelian localization can be used to express the partition function of Chern-Simons theory in terms of the equivariant cohomology of the moduli space of flat connections on M. Here we extend this result to apply to the expectation values of Wilson loop operators which wrap the circle fibers of M. Under localization, such a Wilson loop operator reduce…

Arnold不变量 $J^-$ 和 $J^+$ 的同调提升

Homological lifts of Arnold invariants $J^-$ and $J^+$

本文为Arnold不变量$J^-$和$J^+$构造同调提升，通过区域同调和圆同调细化Viro欧拉积分多项式与量子曲率多项式，并给出严格性证明和同调面积态和模型。

Lee类的可除性及其与Rasmussen不变量的关系

Divisibility of Lee's class and its relation with Rasmussen's invariant

该研究定义了Lee类的$c$-可除度，证明由此构造的$\bar{s}_c$是链环不变量，并通过多方面证据探讨其与Rasmussen的$s$-不变量的一致性。