arXiv research

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 foliation…

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 funda…

The paper studies slopes for knot fillings with left-orderable fundamental groups.

Study shows surgeries on certain knots result in non-left-orderable 3-manifold groups.

Constructs foliations from manifold group left orders.

3-manifolds with Heegaard 2 admit taut foliations if their fundamental group is left-orderable.

New method detects left-orderable surgeries on knot 6_2.

New property helps show many knot fillings are not left-orderable.

The paper develops a method to construct left-orders on homology spheres from Dehn fillings.

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…

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 …

We show that a class of 3-manifolds with non left-orderable fundamental group are Heegaard Floer homology L-spaces

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 $0\le r \le 4$.

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…

The paper finds left orderable slopes for specific double twist knots.

The paper extends techniques to create non-left-orderable manifolds from links with multiple boundary components.

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 yi…

Study slopes on knot manifolds to understand their fundamental groups.

Left orderability proven for certain 3-manifolds with specific foliations.

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 …

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 $0\le r\le 4$.

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.

New description of $(1,1)$ L-space knots leads to non-left-orderable surgeries.

Let $Y$ be a closed, connected, orientable three-manifold admitting a genus one open book decomposition with one boundary component. We prove that if $Y$ is an L-space, then the fundamental group of $Y$ is not left-orderable. This answers a question posed by John Baldwin.

Study determines left-orderable properties of knot covers.

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

Toroidal 3-manifolds have special group structures that can be shown through specific covers.

Study on pretzel knots and their left-orderable properties.

We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.

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, wi…

Study shows surgeries on certain knots yield left-orderable 3-manifolds.

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

The study explores knots and manifolds, proving properties and non-left-orderable 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, orient…

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 w…

Study proves non-left-orderability of 3-manifolds derived from specific 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 equat…

New conditions for circular orderability of direct products, linking to left-orderability of groups.

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.

Study on left orderability of specific knot covers.

New proof shows left orderability of 3-manifold groups with specific foliations.

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…

Study of knot surgeries and JSJ decompositions to tackle $L$-space conjecture.

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.

The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.

This paper completes proofs for left orderable slopes of double twist knots.

Study shows non-left-orderable surgeries for specific L-space knots.

The paper constructs paths of SL2(R) representations for pretzel knots and shows left-orderability conditions.