Product Of Dehn Twists Research Articles

Partial fiber sum decompositions and signatures of Lefschetz fibrations

In his Ph.D. thesis, Burak Ozbagci described an algorithm for computing signatures of Lefschetz fibrations where the input is a factorization of the monodromy into a product of Dehn twists. In this note, we give a reformulation of Ozbagci's algorithm which becomes much easier to implement. Our main tool is Wall's non-additivity formula applied to what we call partial fiber sum decomposition of a Lefschetz fibration over the 2-disk. We show that our algorithm works for bordered Lefschetz fibrations over disk and it yields a formula for the signature of branched covers where the branched loci are regular fibers. As an application, we give the explicit monodromy factorization of a Lefschetz fibration over disk whose total space has arbitrarily large positive signature for any positive fiber genus.

Non-geometric Veering Triangulations

Recently, Ian Agol introduced a class of “veering” ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are “geometric”, i.e. realized in the complete hyperbolic metric with all tetrahedra positively oriented. This article describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra.

Asymptotic spectral flow for Dirac operators of disjoint Dehn twists

Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D + cl(− ir 2 a) as r → ∞. If a is the Thurston–Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is O(r).

Teichmüller disks and hyperbolic mapping classes

Let $\tilde{S}$ be an analytically finite Riemann surface of type $(p,n)$ with $3p+n>3$. Let $x\in \tilde{S}$ and $S=\tilde{S}\backslash \{x\}$. Let $\mbox{Mod}_S^x$ denote the $x$-pointed mapping class group of $S$ and $\mbox{Mod}_{\tilde{S}}$ the mapping class group of $\tilde{S}$. Then the natural projection $J:T(S)\rightarrow T(\tilde{S})$ between Teichmüller spaces induces a group epimorphism $I:\mbox{Mod}_S^x\rightarrow \mbox{Mod}_{\tilde{S}}$. It is well known that for a given Teichmüller disk $\tilde{\Delta}$ in $T(\tilde{S})$, there is a family $\mathscr{F}(\tilde{\Delta})$ of Teichmüller disks $\Delta(z)$ in $T(S)$ parametrized by a hyperbolic plane. If $\tilde{\Delta}$ is invariant under a hyperbolic mapping class $\tilde{\theta}$, then all known hyperbolic mapping classes $\theta\in \mbox{Mod}_S^x$ for which $I(\theta)=\tilde{\theta}$ stem from the construction of $\mathscr{F}(\tilde{\Delta})$. We show that if $\tilde{\theta}$ is represented by a product of Dehn twists along two filling simple closed geodesics, then there exist infinitely many hyperbolic mapping classes $\gamma\in \mbox{Mod}_S^x$ with $I(\gamma)=\tilde{\theta}$ so that their invariant Teichmüller disks are not members of $\mathscr{F}(\tilde{\Delta})$. The result contrasts with the original pattern established by I. Kra.

Crosscap slides and the level 2 mapping class group of a nonorientable surface

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a closed nonorientable surface N generated by all crosscap slides is equal to the level 2 subgroup consisting of those mapping classes which act trivially on H_1(N;Z_2). We also prove that this subgroup is generated by involutions.

Pseudo-Anosov mapping classes and their representations by products of two Dehn twists

Let \( \tilde S \) be a Riemann surface of analytically finite type (p, n) with 3p − 3 + n > 0. Let a ∈ \( \tilde S \) and S = \( \tilde S \) − {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on \( \tilde S \) and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on \( \tilde S \), there are infinitely many pseudo-Anosov maps F on S − {b} = \( \tilde S \) − {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.

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}.

Periodic automorphisms of surfaces and cobordism

In this paper, we work in the differential category. Unless otherwise stated, a surface is an oriented closed, possibly disconnected, surface, and an automorphism is an orientation preserving self-homeomorphism. An automorphism of a surface (/%/) is said to be null-cobordant if there is a compact oriented 3-manifold M equipped with an automorphism (M,/), such that d{M,f) = {dM,f\dM) is equal to {F,f). We call this 3-manifold M the null-cobordism for {F,f). Two automorphisms of surfaces {Fufx) and {F2,f2) are cobordant if {F^f^ { — F2,f2) is null-cobordant. The cobordism classes form a group Δ2+ whose group law is induced by disjoint sum II . Bonahon [B], Edmonds and Eving [EE] proved that Δ 2 + is isomorphic to Z z φ(Z/2Z) z . Bonahon asked the following question in his paper [B section 9] Given an automorphism of a surface {for instance presented as a product of Dehn twists), decide whether it is null-cobordant or not. For the sake of characterizing null-cobordant automorphisms, we want to know, for arbitrary null-cobordant automorphism, what kind of 3-manifold can be constructed as its null-cobordism, and we want to get an explicitly constructed family of 3-manifolds in which, for any null-cobordant automorphism, we can find a null-cobordism of this automorphism. For example, if an automorphism of a 2-torus is null-cobordant then it bounds an automorphism of a solid torus ([B]). In this paper, we show that the same kind of things are true for other surfaces: