arXiv research

Boundary Dehn twists become trivial after abelianization.

Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.

The paper uses Floer homology to study twist coefficients and their behavior after capping off.

Study shows Dehn twist coefficients are consistent across different actions on surfaces.

A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In this paper, we give various results about such collections of curves and the assoc…

Exotic diffeomorphisms found on complex surfaces and 4-manifolds.

Study shows exotic Dehn twists on certain 3-sphere fillings.

Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.

Nontrivial boundary Dehn twist found on K3#K3 manifold.

We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibrat…

For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realiz…

Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.

We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book d…

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fiber…

Johnson kernel generated by specific Dehn twists on surfaces.

The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.

New coefficient detects irrational rotation behavior on infinite-type surfaces.

Researchers compute twisted Reidemeister torsion for hyperbolic 3-manifolds.

By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the…

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\geq 1$ and $k \geq 1$, we show that the $2n$-dimensional Weinstein domain $W = \{f=δ\} \cap B^{2n+2}$, determined by the degree $k$ homogeneous polynomial $f\in \mathbb{C}[z_0,\dots,z_n]$, has a Boothby-Wang type boundary …

Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.

On a compact oriented surface of genus $g$ with $n\geq 1$ boundary components, $δ_1, δ_2,\ldots, δ_n$, we consider positive factorizations of the boundary multitwist $t_{δ_1} t_{δ_2} \cdots t_{δ_n}$, where $t_{δ_i}$ is the positive Dehn twist about the boundary $δ_i$. We prove that for $g\geq 3$, the boundary multitwis…

Example shows smooth vs topological isotopy in a 4-manifold.

Study finds the order of Dehn twists in various groups.

Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties o…

Study connects surface twists to curve invariants.

Paper examines Dehn twists on non-orientable surfaces and their limitations.

Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class…

Study of Dehn twists in free groups generates right-angled Artin groups.

The paper explores methods to decompose periodic maps into Dehn twists.

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

Generalized Dehn twists in surfaces and 3D relate to homology and skein algebras.

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 …

The study examines subgroups of torus mapping class group generated by Dehn twists powers.

It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.

Generates spin structure stabilizers using Dehn twists.

Calculates Dehn twist actions on conformal blocks for modular categories.

We construct nontrivial roots of Dehn twists about nonseparating curves.

We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hiros…

We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h-2 is the only universal bound on the self-intersection number of a section of any such genus g bun…

The study examines Heegaard splittings defined by Dehn twists and finds hyperbolic metrics with specific geodesic lengths.

Positive factorization for pseudoperiodic homeomorphisms on surfaces.

Presentations for involutions on non-orientable surfaces up to genus 5.

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of ra…

Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.