Bradley Meyer, Anh T. Tran
We compute the nonabelian $\mathrm{SL_2}(\mathbb{C})$-character varieties of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m$ and $n$ are non-zero integers. By studying real points on these varieties, we determine the left orderability of the fundamental groups of the cyclic branched covers of $C(2n+1,2m,2)$.
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We prove the non-left-orderability of the fundamental group of the $n$-th fold cyclic branched cover of the pretzel knot $P(3,-3,-2k-1)$ for all integers $k$ and $n\ge 1$. These $3$-manifolds are $L$-spaces discovered by Issa and Turner.
We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
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.
For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these knots is left-orderable. There are knots, including the figure-eight knot, for which this set is empty. We give the first class of hyperbolic knots, not of this type, for which these integers can be completely determined.
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.
We construct a 1-parameter family of $\mathrm{SL}_2(\mathbf{R})$ representations of the pretzel knot $P(-2,3,7)$. As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.
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…
Motivated by Clay and Watson's question on left-orderability of the fundamental group of the resultant space of an $r'$-surgery on the $(p, q)$-cable knots for $r' \in (pq-p-q,pq)$, this paper proves by elementary means that for specific pairs of $(p,q)$-cable knots of torus knots, $r' \in [pq-1,pq]$ gives a surgery yielding non-left orderable fundamental groups.
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 $0$.
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.
Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
本文利用 Mayland 技术证明若干亏格一及高亏格环面链环结的纽结群双可序，首次给出非纤维化或非交错纽结的双可序例子。
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.
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 $n > 2π/ \arccos(1-2/(1+a_1 a_2 + a_2 a_3 + a_3 a_1))$.
We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 599 of the corresponding knot groups. With our methods we are able to deal with 191 more.
Orderability of link quandles
本文发展了quandles可序性的一般理论，证明许多纤维素素数纽结的纽结quandles右可序，而多数非平凡环面链环的链环quandles不可右序，并探讨了其与包络群可序性的联系。
A combinatorial approach to obstructing left-orderability of Dehn fillings
该研究引入结的共轭斜率性质，用于研究德恩填充产生的基本群的非左可序性，证明其对特定结成立，此性质还限制结群左序类型，并表明子午线在许多结的德恩填充基本群中的像为广义挠元。
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.
We show that any exceptional non-trivial Dehn surgery on a hyperbolic two-bridge knot yields a 3-manifold whose fundamental group is left-orderable. This gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
Chainmail links and non-left-orderability
本文证明了平坦全增广链甲链环的交错手术产生全L空间，并利用序检测研究了与L空间结连通和手术的非左序性。
We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
We prove that the knot groups of $6_2$ and $7_6$ are not bi-orderable. These are the only two knot groups up to 7 crossings whose bi-orderability was not known. Our method applies to a very broad class of knots.
We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3-sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be applied for a finitely generated group which is an HNN extension with certain properties.
Let L \subset S^3 denote an alternating link and Sigma(L) its branched double-cover. We give a short proof of the fact that the fundamental group of Sigma(L) admits a left-ordering iff L is an unlink. This result is originally due to Boyer-Gordon-Watson.
Knot groups, quandle extensions and orderability
本文利用quandle可序性刻画L空间3维流形，并研究quandle动力学扩张及链环quandle的右可序与右循环可序性。
A branched covering surface-knot is a surface-knot in the form of a branched covering over a surface-knot. For a branched covering surface-knot, we have a numerical invariant called the simplifying number. We show that branched covering surface-knots with degree three have the simplifying numbers less than three.
For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.
We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.
For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.
Characterising knots by their branched coverings
本文通过循环分支覆盖方法研究结的表征问题，探讨其在结分类中的应用与贡献。
We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.
In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fold branched cover is a rational homology sphere. In addition, when K is a 2-bridge knot, we prove that the knot Floer homology of the lifted knot in a particular Spin^c structure in the branched double cover matches the knot Floer homology of the original knot K in S^3. We conclud…
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $Σ_m(K)$ obtained from a grid diagram for $K$, we give a combinatorial proof for the invariance of the associated combinatorial knot Floer homology over $\mathbb{Z}$.
A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications.
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $Σ_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$.
Transverse knots determined by their cyclic branched covers
本文研究横截结的横截同胚类，通过构造其他非同胚但循环分支覆盖接触同胚的横截结例子，证明许多横截结的横截同胚类由其循环分支覆盖的接触同胚类型决定。
A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications for the case when branched covering surface-knots have a non-zero number of bra…
In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.
A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. To prove the assertion, we give a presentation of the fundamental group of the complement of a branched twist spin, which generalizes a presentation of Plotnick, calculate the first elementary ideals and obtain the condition of the knot determinants by substituting $-1$ for t…
An $(m,n)$-branched twist spin is a fibered $2$-knot in $S^4$ which is determined by a $1$-knot $K$ and coprime integers $m$ and $n$. For a $1$-knot, Lin proved that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of a $1$-knot up to conjugation is determined by the knot determinant of the $1$-knot. In this paper, we prove that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of an $(m,n)$-branched twist spin up to conjugation is determined by the determinant of a $1$-knot in the orbit space by comparing a prese…
Strongly-cyclic branched coverings of knots are studied by using their (g,1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations.
We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.
We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1,1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic pre…
We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, which are cyclic presentations in the case of 2-bridge knots. The homology groups are given for a wide class of cases. Moreover, we prove that each singly-cyclic branched covering of a 2-bridge link is the composition of a meridian-cyclic branched covering of a determined link and…
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic 2-fold branched covering of a knot in the 3-sphere. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the st…
We show that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a consequence, we obtain a parametrization of (1,1)-knots by 4-tuples of integers. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice punctured torus, we provide an algorithm which gives the parametrization of all torus knots.
Residual torsion-free nilpotence has proven to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of parafree groups. This paper proves Mayland's assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link gro…
By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us to characterise invertibility and other properties in the hyperbolic case. Moreover, with our construction we are able to detect both the trivial knotoid in the 2-sphere and the trivial planar knotoid.
Folding Branched Covers of the $3$-Sphere Branched over Knots
本文研究了3-球面在纽结上分支的分支覆盖的折叠映射，证明了循环和二面体分支覆盖均可被折叠，并详细探讨了沿着(2,5)型环面纽结分支的二面体覆盖的折叠。
The groups of branched twist-spun knots
该研究根据3-流形群刻画经典纽结的分支扭转旋的群，给出基于$PD_3$-群的代数猜想刻画，并证明每个群对应有限个分支扭转旋。
We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted version of the toroidal grid diagrams recently introduced by Manolescu, Ozsvath, and Sarkar. We conclude with a discussion of how one might obtain nice Heegaard diagrams for cyclic branched covers of more general knots.
We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicit some fundamental differences between these two representations. For instance, seeing the figure eight knot complement as a surface bundle over the circle, the behaviour of of the fundamental group of the fiber under the representation is a key difference between rho_1 and rho_2.
Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$ such that its links are the total spaces of the $n$-branched cyclic coverings over the corresponding knots. In this way we obtain as links of singularities, $3$-manifolds that does not appear in the complex case, such as hyperbolic $3$-manifolds or the Hantzsche-Wendt manifold.
A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two--fold branched co…
We show that the resulting manifold by $r$-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope $r$ satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates $L$-spaces and the left-orderability of their fundamental groups.
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\bf S^3$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $Υ_m^C (K)$ of knots in $S^3$. We give computations of these invariants for some families of alternating knots and reprove some independence results in the smooth concordance group.
The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along branched twist spins are studied by Fintushel, Gordon and Pao. In this paper, we determine the 2-knot obtained from $K$ by the Gluck twist along $K^{\prime}$. As an application, we give infinitely many pairs of inequivalent branched twist spins whose complements are homeomorphic…
Distinguishing 2-knots admitting circle actions by fundamental groups
本文研究3-球面中不同1-结与互质整数对构造的分支扭转旋转的唯一性问题，证明大多数情况下不存在产生相同分支扭转旋转的不同1-结，并利用循环型3-轨道流形的基本群作为补空间基本群的商群来区分分支扭转旋转。
A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficients and 2) the spin^c structures are more readily discernable. We apply this result to a branch locus which is a braid, and use the braid structure to find information about a fibered knot in the branched double cover. In some cases this provides all the information about the knot …
Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. This paper shows that the knot Floer homology of this lift, with mod 2 coefficients, can be computed from a spectral sequence starting at a type of Khovanov homology already described by Asaeda, Przytycki, and Sikora. We extend the known results about this type of Khovanov homology,…
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