Nonseparating Curve Research Articles

Finite rigid sets of the non-separating curve complex

Abstract We prove that the non-separating curve complex of every surface of finite type and genus at least three admits an exhaustion by finite rigid sets.

On Squares of DEHN Twists About Non-separating Curves of A Non-orientable Closed Surface

The level 2 mapping class group of an orientable closed surface can be generated by squares of Dehn twists about non-separating curves (see [5]). On the other hand, the level 2 mapping class group $$\mathcal {M}_2(N)$$ of a non-orientable closed surface N can not be generated by only Dehn twists, and so it can not be generated by squares of Dehn twists about non-separating curves. In this paper, we prove that the Dehn twist subgroup of $$\mathcal {M}_2(N)$$ can not be generated by squares of Dehn twists about non-separating curves either. As an application, we give a finite generating set for the subgroup of $$\mathcal {M}_2(N)$$ generated by Dehn twists about separating curves and squares of Dehn twists about non-separating curves. Moreover, we examine about actions on non-separating simple closed curves of N by $$\mathcal {M}_2(N)$$ .

Stable specific torsion length and periodic mapping classes

We show that for any periodic mapping class, there is some power which maps a nonseparating, simple closed curve to a distinct, disjoint nonseparating curve. As an application of this result, we introduce the notion of stable specific torsion length of a group element and show that the stable specific torsion length of a Dehn twist is bounded above by six.

Experimental Statistics for Mirzakhani’s Theorem

In her seminal 2008 article, Maryam Mirzakhani showed that the ratio that two topological types of curves occur in is a rational number. In this article, we describe the process by which we obtained experimental evidence that separating and non-separating curves on the surface of genus two occur in the ratio . 1

Edge-preserving maps of the nonseparating curve graphs, curve graphs and rectangle preserving maps of the Hatcher–Thurston graphs

Let [Formula: see text] be a compact, connected, orientable surface of genus [Formula: see text] with [Formula: see text] boundary components with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the nonseparating curve graph, [Formula: see text] be the curve graph and [Formula: see text] be the Hatcher–Thurston graph of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is closed and [Formula: see text] is a rectangle preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We also prove that these homeomorphisms are unique up to isotopy when [Formula: see text].

Geometry of the graphs of nonseparating curves: covers and boundaries

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenstädt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries.

Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona-Valdez, this proves also that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic.

Signatures of surface bundles and scl of a Dehn twist

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second one gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third one has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bounds for stable commutator lengths of Dehn twists by giving factorizations of powers of Dehn twists as products of commutators. One of the factorizations is used to construct the second examples of surface bundles. As a corollary, we see that there is a gap between the stable commutator length of the Dehn twist along a nonseparating curve in the mapping class group and that in the hyperelliptic mapping class group if the genus of the surface is greater than or equal to 8.

The Neuwirth Conjecture for a family of satellite knots

Let [Formula: see text] be a nontrivial knot in [Formula: see text]. It was conjectured that there exists a Neuwirth surface for [Formula: see text]. That is, a closed surface in [Formula: see text] containing the knot [Formula: see text] as a nonseparating curve and such that every compressing disk for the surface intersects the knot in at least two points. We provide explicit constructions of Neuwirth surfaces for a family of satellite knots, which do not depend on the existence of nonorientable algebraically incompressible and [Formula: see text]-incompressible spanning surfaces for these knots.

Rigidity of the nonseparating and outer curve graph

Let $$S_{1}$$ and $$S_{2}$$ be orientable surfaces of finite topological type with empty boundary, both of genus at least 1 and $$n_{1}, n_{2} \ge 0$$ punctures. We define the nonseparating and outer curve graph $$\mathcal {NO}(S_{i})$$: a particular induced subgraph of the curve graph, which has the same large-scale geometry. We prove (under certain conditions on the complexity of $$S_{1}$$ and $$S_{2}$$) that if $$\varphi : \mathcal {NO}(S_{1}) \rightarrow \mathcal {NO}(S_{2})$$ is a superinjective map, then $$S_{1}$$ is homeomorphic to $$S_{2}$$ and $$\varphi $$ is induced by a homeomorphism.

Positive factorizations of mapping classes

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

Automorphisms groups of simplicial complexes of infinite type surfaces

Let S be an orientable surface of innite genus with a nite numberof boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S), and the Schmutz graph G(S) of S. When all topological ends of S carry genus, we show that all elements in the automorphismgroups Aut(C(S)), Aut(N(S)), and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG(S) of the innite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S)).

Topological complexity of symplectic 4-manifolds and Stein fillings

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by Auroux, Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci and Stipsicz, independently.

On roots of Dehn twists

Margalit and Schleimer \cite{MS} discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. We give a complete set of conjugacy invariants for such a root by using a classification theorem of Matsumoto and Montesinos \cite{MM1, MM2} for pseudo-periodic maps of negative twists. As an application, we determine the range of degree for roots of a Dehn twist.

FRACTIONAL POWERS OF DEHN TWISTS ABOUT NONSEPARATING CURVES

AbstractLet Sg be a closed orientable surface of genus g ≥ 2 and C a simple closed nonseparating curve in F. Let tC denote a left-handed Dehn twist about C. A fractional power of tC of exponent ℓ//n is an h ∈ Mod(Sg) such that hn = tCℓ. Unlike a root of a tC, a fractional power h can exchange the sides of C. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if gcd(ℓ,n) = 1, then h will be isotopic to the ℓth power of an nth root of tC and that n ≤ 2g+1. In general, we show that n ≤ 4g, and that side-preserving fractional powers of exponents 2g//2g+2 and 2g//4g always exist. For a side-exchanging fractional power of exponent ℓ//2n, we show that 2n ≥ 2g+2, and that side-exchanging fractional powers of exponent 2g+2//4g+2 and 4g+1//4g+2 always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on S5.

Degree of roots of disk twists on 3-dimensional handlebodies

Margalit and Schleimer (Geom Topol 13(3):1495–1497, 2009) discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. McCullough and Rajeevsarathy (Geom Dedicata 151(1):397–409, 2011) and Monden (Rocky Mt J Math, to appear) obtained the evaluation of the degrees of roots of Dehn twists. In this paper, we discuss existence and degrees of homeomorphisms whose power is equal to disk twist about a nonseparating disk in the mapping class group of the 3-dimensional handlebody.

$U$-action on perturbed Heegaard–Floer homology

This paper has two purposes. First, as a continuation of Perturbed Floer homology of some fibered three manifolds, we apply a similar method to compute the perturbed HF^+ for some special classes of fibered three-manifolds in the second highest spin^c_- structures, including the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves. Second, we establish an adjunction inequality for the perturbed Heegaard Floer homology, which indicates a potential connection between the U-action on the homology group and the Thurston norm of a three-manifold. As an application, we find the U-action on the perturbed HF^+ of the above classes of fibered three-manifolds is trivial.

Roots of Dehn twists

Margalit and Schleimer found examples of roots of the Dehn twist t C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h n = t C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n th root of t C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, n th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n th roots having n greater than or equal to the genus.

Dehn twists have roots

We construct nontrivial roots of Dehn twists about nonseparating curves.

Tight contact structures and genus one fibered knots

We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual non-separating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the corresponding contact structure is tight or overtwisted. We rely on Ozsv{\'a}th-Szab{\'o} Heegaard Floer homology in our construction and, in particular, we completely identify the $L$-spaces with genus one, one boundary component, pseudo-Anosov open book decompositions. Lastly, we reveal a new infinite family of hyperbolic three-manifolds with no co-orientable taut foliations, extending the family discovered in \cite{RSS}.