arXiv research

Paper classifies $CFK^\infty$ type of almost L-space knots.

New proof shows certain 3D shapes can't be instanton L-spaces.

Classifies 3-braids as L-space knots.

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

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

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

The paper explores nonexistence and existence of symplectic and Stein fillable contact structures on 3-manifolds.

Proves left-orderability of certain Dehn fillings on 3-manifolds.

Study shows inequality in Floer homologies for 3-manifold covers.

The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obt…

Study shows 3D shapes can't fit in certain 4D spaces.

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

Study proves non-left-orderability of 3-manifolds derived from specific knots.

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…

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examp…

We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the t…

We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorith…

Classifies SU(2)-abelian graph manifolds with a single JSJ torus.

The paper characterizes L-space 3-manifolds using quandle orderability.

The paper finds infinite families of non-left-orderable L-spaces from tangles.

Study positive braid knots and their taut foliations, proving some L-space conjecture evidence.

Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.

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…

The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

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 $K$.

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeri…

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

Enhances knot surgery formulae for instanton Floer homology.

New foliations show knot meridians are detectable.

Computes upper bounds for instanton knot homology.

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, Lewalle…

Paper tackles $L$-space conjecture for knot manifolds, proving equivalence for some properties.

New contact structures found on Brieskorn spheres.

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…

Study surgeries on Klein bottle knots in 3-manifolds using Heegaard Floer homology.

The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.

Study shows hyperbolic knots' monodromy without fixed points.

If a 3--manifold $Y$ contains a non-separating sphere, then some twisted Heegaard Floer homology of $Y$ is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in $L$--spaces.

Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when $n\!=\!1$ and $\mathcal{L}$ is nontrivial. For $n\mkern-2mu>\mkern-3mu1$, howeve…

Study Froyshov invariants and closed geodesics in hyperbolic 3-manifolds.

Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.

New method constructs actions on R capturing foliations, proving left-orderability.

We show that any co-orientable foliation of dimension two on a closed orientable $3$-manifold with continuous tangent plane field can be $C^0$-approximated by both positive and negative contact structures unless all the leaves are simply connected. As applications we deduce that the existence of a taut $C^0$-foliation …

(1,1) non-L-space knots are foliar in 3D space.

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits very negative (i.e. $d_{1}, d_{2}\ll0$) L-space surgeries, it is the Hopf link. We also give a w…

We give some remarks on two closely related issues as stated in the title. In particular we show that a Montesinos knot is SU(2)-simple if and only if it is a 2-bridge knot, extending a result of Zentner for 3-tangle summand pretzel knots. We conjecture with some evidence that an SU(2)-cyclic rational homology 3-sphere…