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Tetsuya Ito

Publications and source records attributed to Tetsuya Ito.

At least 55 records · Page 3Linked to original sources
arXiv2019

Nontrivial elements in a knot group which are trivialized by Dehn fillings

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K) coincides with that of the inverse of u, and u and its inverse u correspond to a slope r, a rational number or 1/0, we write << r >> = << u >>. The normal closure << u >> describes elements which are trivialized by r-Dehn filling of E(K). In this article, we prove that << r_1 >> =<< r_2 >> if and only if r_1 = r_2, and for a given finite family of slopes S = {r_1, ..., r_n}, the intersection of << r_1 >> , << r_2>>, ..., and << r_n >> contains infinitely many elements except when K is a (p, q)-torus knot and pq belongs to S. We also investigate inclusion relation among normal closures of slope elements.

math.GT↗
arXiv2019

Chirally cosmetic surgeries and Casson invariants

We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariants. As applications, we give a complete classification of chirally cosmetic surgeries on alternating knots of genus one.

math.GT↗
arXiv2018

Generalized torsion and decomposition of 3-manifolds

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $3$-manifold $M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $3$-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements.

math.GT↗
arXiv2018

Positivities of knots and links and the defect of Bennequin inequality

We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of Bennequin inequality. Finally, we determine strong quasipositivity and quasipositivity for knots up to 12 crossings (with two exceptions for quasipositivity).

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arXiv2018

Quasi right-veering braids and non-loose links

We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link $K$ in a contact 3-manifold $(M,ξ)$ is non-loose if and only if every braid representative of $K$ with respect to every open book decomposition that supports $(M,ξ)$ is quasi-right-veering. We also show that several definitions of "right-veering" closed braids are equivalent.

math.GT↗
arXiv2018

On the fractional Dehn twist coefficients of branched coverings

We discuss how the fractional Dehn twist coefficient behaves under a fully ramified branched covering of an open book, and give applications to both topological and contact 3-manifolds. Among them, we show that non-right-veering closed braids represent virtually loose transverse links.

math.GT↗
arXiv2017

The defect of Bennequin-Eliashberg inequality and Bennequin surfaces

For a null-homologous transverse link $\mathcal T$ in a general contact manifold with an open book, we explore strongly quasipositive braids and Bennequin surfaces. We define the defect $δ(\mathcal T)$ of the Bennequin-Eliashberg inequality. We study relations between $δ(\mathcal T)$ and minimal genus Bennequin surfaces of $\mathcal T$. In particular, in the disk open book case, under some large fractional Dehn twist coefficient assumption, we show that $δ(\mathcal T)=N$ if and only if $\mathcal T$ is the boundary of a Bennequin surface with exactly $N$ negatively twisted bands. That is, the Bennequin inequality is sharp if and only if it is the closure of a strongly quasipositive braid.

math.GT↗
arXiv2017

Positive factorizations of symmetric mapping classes

We study Question 7.9 in the paper "Monoids in the mapping class group" by Etnyre and Van Horn-Morris; whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid. We answer affirmatively for mapping classes satisfying certain cyclic conditions.

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arXiv2015

Essential open book foliation and fractional Dehn twist coefficient

We introduce an essential open book foliation, a refinement of the open book foliation, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the `gap' of overtwisted contact structures and a non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen-Thurston types of the monodromies of their open book decompositions.

math.GT↗
arXiv2015

Coverings of open books

We study a coverings of open books and virtually overtwisted contact manifolds using open book foliations. We show that open book coverings produces interesting examples such as transverse knots with depth grater than 1. We also demonstrate explicit examples of virtually overtwisted open books.

math.GT↗
arXiv2015

On a structure of random open books and closed braids

A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.

math.GT↗