On L-spaces and non left-orderable 3-manifold groups

Integral left-orderable surgeries on genus one fibered knots

Following the classification of genus one fibered knots in lens spaces by Baker, we determine hyperbolic genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3-manifolds with left-orderable fundamental groups.

An explicit description of $(1,1)$ L-space knots, and non-left-orderable surgeries

Greene, Lewallen and Vafaee characterized $(1,1)$ L-space knots in $S^3$ and lens space in the notation of coherent reduced $(1,1)$-diagrams. We analyze these diagrams, and deduce an explicit description of these knots. With the new description, we prove that any L-space obtained by Dehn surgery on a $(1,1)$-knot in $S^3$ has non-left-orderable fundamental group.

Non left-orderable surgeries on L-space twisted torus knots

We show that if $K$ is an L-space twisted torus knot $T^{l,m}_{p,pk \pm 1}$ with $p \ge 2$, $k \ge 1$, $m \ge 1$ and $1 \le l \le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever $\frac{r}{s} \ge 2 g(K) -1$, where $g(K)$ is the genus of $K$.

左可序与非左可序HNN扩张的例子

Examples of left-orderable and non-left-orderable HNN extensions

本文证明了 torsion-free nilpotent 群的 HNN 扩张是左可序的，并构造了非左可序的 HNN 扩张例子。

左可序性与可定向紧致叶面

Left orderability and taut foliations with orderable cataclysm

本文研究了3维流形的左可序性，通过可定向紧致叶面和伪Anosov流的性质，证明了特定条件下的流形基群的左可序性，并探讨了Dehn填充的应用。

On left-orderable fundamental groups and Dehn surgeries on twist knots

We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$ is even, and $r \in [0,4] \cup (\frac{4m}ω+4, 2m+4)$ if $m$ is odd, where $ω>1$ is the unique real solution of the equation $x e^{x}=4m$.

Infinite families of non-left-orderable L-spaces

For each connected alternating tangle, we provide an infinite family of non-left-orderable L-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational homology 3-sphere is an L-space if and only if it is non-left-orderable. These 3-manifolds are obtained as Dehn fillings of the double branched covering of any alternating encircled tangle. We give a presentation of these non-left-orderable L-spaces as double branched coverings of S^3, branched over some specified links that turn out to be hyperbolic. We show that the obtained families include many non…

链环上的序检测与非左序Dehn填充

Order-detection and non-left-orderable surgeries on links

本文利用斜率的序检测技术，将生成不可左序基本群Dehn填充的计算方法推广至多环面边界3-流形，并构造双曲链环无限族验证了L-space猜想。

On $1$-bridge braids, satellite knots, the manifold $v2503$ and non-left-orderable surgeries and fillings

We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot $K$ with property (D) with slope $\frac{p}{q}\ge 2g(K)-1$ is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive $1$-bridge braids have property (D); (2) L-space satellite knots, with positive $1$-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on $v2503$ is not left orderable. Additionally…

Slope of Orderable Dehn Filling of Two-Bridge Knots

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.

环面三维流形具有可循环排序的基本群

Toroidal 3-manifolds have circularly orderable fundamental groups

研究环面三维流形基本群的可循环排序问题，通过证明其存在有限循环覆盖且基本群可左排序来实现，还验证了相关猜想，揭示了环面三维流形与L-空间猜想等的关系。

通过双曲 $\widetilde{PSL}(2,\mathbb{R})$-表示研究纽结 $6_2$ 上的左序手术

Left-orderable surgeries on the knot $6_2$ via hyperbolic $\widetilde{PSL}(2,\mathbb{R})$-representations

提出一种检测纽结 Dehn 手术基本群左序性的新方法，并通过双曲 $\widetilde{PSL}(2,\mathbb{R})$-表示证明了纽结 $6_2$ 在特定斜率区间内的所有手术均具有左序基本群。

Orderable 3-manifold groups

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds' groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, …

双扭结的左序手术 II

Left orderable surgeries of double twist knots II

本文完善了双桥纽结左序斜率存在性的证明，利用双曲SL₂(ℝ)表示的连续族，证明C(2m+1,-2n)在区间(-4n,4m)内所有斜率均为左序斜率。

关于链环的对合广群的左可序性

On left-orderability of involutory quandles of links

该研究提出非分裂链环对合广群的非左可序性判据，改进了已有定理，证明多类链环的对合广群非左可序，并提出拟交错链环相关猜想。

Left-orderable fundamental group and Dehn surgery on twist knots

For any hyperbolic twist knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ satisfies the inequality $0\le r \le 4$.

Left-orderable fundamental group and Dehn surgery on the knot $5_2$

We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number 3/7, has left-orderable fundamental group if the slope $r$ satisfies $0\le r\le 4$.

$\mathrm{SL}_2(\mathbb R)$-表示与$(-2, 3, 2n+1)$-pretzel结的左可排序手术

$\mathrm{SL}_2(\mathbb R)$-representations and left-orderable surgeries of $(-2, 3, 2n+1)$-pretzel knots

本文通过构造连续的$\mathrm{SL}_2(\mathbb R)$-表示，研究了$(-2, 3, 2n+1)$-pretzel结的群表示，并证明了特定条件下所得3-流形的fundamental group是左可排序的。

Left-orderable fundamental group and Dehn surgery on genus one two-bridge knots

For any hyperbolic genus one 2-bridge knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ lies in some range which depends on the knot.

检测到的Seifert曲面与左可排序手术的区间

Detected Seifert surfaces and intervals of left-orderable surgeries

本文受$L$-空间猜想启发，利用Culler-Dunfield技术构造表示弧检测Seifert曲面，结合Gao的全纯延拓轨迹方法，证明积分同调实心环上0附近Dehn填充的左可排序性，并验证无限奇pretzel纽结的区间，为新的无限闭3流形族证实猜想。

Left-orderable fundamental groups and Dehn surgery

There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliations. As a result, many large classes of 3-manifolds, including knot complements, are known to have left-orderable fundamental group. However, though the complement of a knot has left-orderable fundamental group, the result of Dehn surgery is a closed 3-manifold that need not have t…

Non-left-orderable double branched coverings

We develop a method to show the fundamental group of the double branched covering of a link is not left-orderable by introducing the notion of the coarse presentation. As in the usual group presentations, a coarse presentation is given by a set of generators and relations, but inequalities are allowed as relations. By using coarse presentation, we give a family of links whose double branched covering has the non-left-orderable fundamental group. Our family of links includes many known examples and new examples.

Non-left-orderable 3-manifold groups

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S^3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S^3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 5_2 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold wi…

Left-orderability for surgeries on twisted torus knots

We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to $0$.

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space …

Left-orderablity for surgeries on $(-2,3,2s+1)$-pretzel knots

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a $(-2,3,2s+1)$-pretzel knot ($s\ge 3$) with slope $\frac{p}{q}$ is not left orderable if $\frac{p}{q}\ge 2s+3$, and that it is left orderable if $\frac{p}{q}$ is in a neighborhood of zero depending on $s$.

Taut foliations in branched cyclic covers and left-orderable groups

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible $3$-manifolds containing a genus $1$ fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology $3$-sphere …

L-spaces, left-orderability and two-bridge knots

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space and its fundamental group is not left-orderable. Therefore the family of 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ verifies the $L$-space conjecture. We also show that if $K_{[2k,-2l]}$ is a 2-bridge knot with $k\geq 2$, $l>0$, then the fundamental group of the 5-fold cyclic branched cover of $K_{[2k,-2l]}$ is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus one two-b…

Wiegold 问题与左可序群的自由积

The Wiegold problem and free products of left-orderable groups

本文通过构造自由积的左序，证明了左可序群的自由积具有正规秩大于1，从而解决了Wiegold问题。

On left-orderability and cyclic branched coverings

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for which the r-th cyclic branched covering of S^3 along the knot is left-orderable.

Non-left-orderable surgeries on twisted torus knots

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in the 3-sphere, it is natural to ask what conditions on the knot group are sufficient to imply that the quotient associated to Dehn surgery is not left-orderable. Clay and Watson develop a criterion for determining the left-orderability of this quotient group and use it to verify the conjecture for surgeries on certain L-space twisted torus knots. We general…

A Note on the Orderability of Dehn Fillings of the Manifold $v2503$

We show that Dehn filling on the manifold $v2503$ results in a non-orderable space for all rational slopes in the interval $(-\infty , -1)$. This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.

Left orderability for surgeries on the $[1,1,2,2,2j]$ two-bridge knots

Let $M$ be a $\mathbb{Q}$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left-orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\geq 1$ admits an interval of left-orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.

Non-Left-Orderable Surgeries on 1-Bridge Braids

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in $S^3$ can produce large families of L-spaces, it is natural to examine the conjecture on these 3-manifolds. Greene, Lewallen, and Vafaee have proved that all 1-bridge braids are L-space knots. In this paper, we consider three families of 1-bridge braids. First we calculate the knot groups and peripheral subgroups. We then verify the conjecture on the three cases by applying the criterion developed by…

On L-spaces and left-orderable fundamental groups

Examples suggest that there is a correspondence between L-spaces and 3-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group …

Classical pretzel knots and left orderability

We consider the classical pretzel knots $P(a_1, a_2, a_3)$, where $a_1, a_2, a_3$ are positive odd integers. By using continuous paths of elliptic $\mathrm{SL}_2(\mathbb R)$-representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$-surgery on $P(a_1, a_2, a_3)$ has left orderable fundamental group if $\frac{m}{l} < 1$, and (ii) the $n^{\mathrm{th}}$-cyclic branched cover of $P(a_1, a_2, a_3)$ has left orderable fundamental group if $n > 2π/ \arccos(1-2/(1+a_1 a_2 + a_2 a_3 + a_3 a_1))$.

Promoting circular-orderability to left-orderability

Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G \times \mathbb{Z}/n\mathbb{Z}$ is circularly-orderable for all $n > 1$. This implies that every circularly-orderable group which is not left-orderable gives rise to a collection of positive integers that exactly encode the obstruction to left-orderability, which we call the obstruction spectrum. We precisely describe the behaviour of the obstruction spectrum with respect…

Left-orderability and cyclic branched coverings

We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^3$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer-Gordon-Watson. As an application of this sufficient condition, we show that for any $(p,q)$ two-bridge knot, with $p\equiv 3 \text{ mod } 4$, there are only finitely many cyclic branched covers whose fundamental groups are not left-orderable. This answers a question posed by D{\c a}bkowski, Przytycki and Togha.

纽补的JSJ分解、Dehn手术与$L$-空间猜想

JSJ decompositions of knot exteriors, Dehn surgery and the $L$-space conjecture

本文应用坡度检测技术研究$L$-空间猜想下卫星纽结Dehn手术所得环面3-流形性质，证明特定纽结外部的JSJ图为有根区间，并得出复合纽结有理手术具左序基本群等结论，为共定向紧叶状结构研究提供新方法。

Slope detection, foliations in graph manifolds, and L-spaces

A graph manifold rational homology $3$-sphere $W$ with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use this to show that there are smooth foliations on the pieces of $W$ which come close to matching up on its JSJ tori. This is applied to prove that a graph manifold with left-orderable fundamental group is not an L-space.

Generalized torsions in once punctured torus bundles

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.

On left-orderability and double branched covers of Kanenobu's knots

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

关于非序流形中切触微分同胚环与勒让德子流形环的拓扑

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

本文研究了非序闭切触流形的切触微分同胚环与勒让德子流形环空间的全局拓扑，通过正性度量进行过滤，并证明了其同伦群为正环子空间同伦群的子群。

可排序Coxeter三维多胞形的实射影形变空间的平滑性

The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes

本文证明，对于可排序且正规类型的Coxeter三维多胞形，若其底层组合多胞形非多边形上的锥，则其形变空间为光滑流形，通过分析到实现空间的自然映射得出。

左可排序性与单侧分支的紧致定向不可约3-流形

Left orderability and taut foliations with one-sided branching

本文研究了具有单侧分支的紧致定向不可约3-流形的左可排序性，证明其基本群是左可排序的。

Order-detection of slopes on the boundaries of knot manifolds

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres obtained by gluing compact manifolds along torus boundary components have left-orderable fundamental groups and when a Dehn filling of a knot manifold has a left-orderable fundamental group. Our developments parallel existing results in the case of Heegaard-Floer slope detection and foliation slope detection, leading to several conjectured structure theorems that connect relative Heegaard-Floer homology a…

有序基、保持顺序的自同构与双可排序的链群

Ordered bases, order-preserving automorphisms and bi-orderable link groups

本文提出了一种准则，证明自由群在特定自同构下可双可排序，并应用该结果证明魔法流形的基本群可双可排序。

实理想点、Conway球面与左可序Dehn填充

Real ideal points, Conway spheres, and left-orderable Dehn fillings

本文研究含本质Conway球面的纽结外部的$SL_2(\boldsymbol{C})$特征簇的实理想点，结合相关结果验证了部分纽结大斜率Dehn填充的$L$空间猜想，还对三类纽结族及28个普查外部完成了相关计算验证。

Strong L-spaces and left orderability

We introduce the notion of a strong L-space, a closed, oriented rational homology 3-sphere whose Heegaard Floer homology can be determined at the chain level. We prove that the fundamental group of a strong L-space is not left-orderable. Examples of strong L-spaces include the double branched covers of alternating links in S^3.

Circular orderability and quandles

In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right (respectively left) orderings of a quandle embeds in its space of right (respectively left) circular orderings. Examples of quandles that are not left circularly orderable and examples of quandles that are neither left nor right circularly orderable are given.

Ordered groups and topology

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begins with an introduction to orderable groups and their algebraic properties. Many of the algebraic results are proved by topological methods, via consideration of the space of orderings. After a discussion Hölder's theorem and some dynamical aspects of orderable groups, we provide explicit orderings of important groups in topology: free groups and most surface…

Circularly ordering direct products and the obstruction to left-orderability

Motivated by the recent result that left-orderability of a group $G$ is intimately connected to circular orderability of direct products $G \times \mathbb{Z}/n\mathbb{Z}$, we provide necessary and sufficient cohomological conditions that such a direct product be circularly orderable. As a consequence of the main theorem, we arrive at a new characterization for the fundamental group of a rational homology 3-sphere to be left-orderable. Our results imply that for mapping class groups of once-punctured surfaces, and other groups whose actions on $S^1$ are cohomologically rigid, the products $G \t…

斜率检测与环形3流形

Slope detection and toroidal 3-manifolds

本文研究环形3流形的斜率检测，证明环形整数同调球体的基群可左序化，并展示素卫星结的循环分支覆盖不是L-空间且基群可左序化，同时证明纤维化伴随的循环分支覆盖具有可定向紧致叶面分解。

3-流形群的循环可序性

Circular orderability of 3-manifold groups

受L-space猜想启发，本文首次研究3-流形群的循环可序性，给出紧致连通$\u2119^2$-不可约3-流形基本群循环可序的等价条件，提出L-space猜想的循环可序版本，证明几乎所有graph manifold基本群循环可序，并对比两种可序性在相关运算下的行为。