Paper finds a minimal set of Dehn twists to generate twist subgroup of non-orientable surfaces.
problem Finding a minimal set of Dehn twists to generate the twist subgroup of non-orientable surfaces.
method Used Dehn twists and an argument from Hirose [5] to find a minimal generating set.
result Found a generating set with one less element than the lower bound.
Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. Study of Dehn twists in free groups generates right-angled Artin groups.
problem Understanding dynamics of Dehn twists in free groups.
method Geometry of spheres, tori, and curves in a doubled handlebody; analysis of compatibility conditions.
result Sufficiently large powers of Dehn twists generate right-angled Artin groups.
Generalized Dehn twists in surfaces and 3D relate to homology and skein algebras.
problem Understanding and classifying Dehn twists in various contexts.
method Algebraic and geometric constructions, including automorphisms of fundamental groups and diffeomorphisms.
result Generalized Dehn twists are realized as diffeomorphisms and have algebraic structures.
The study examines subgroups of torus mapping class group generated by Dehn twists powers.
problem Characterizing subgroups generated by powers of Dehn twists.
method Using the ping pong lemma and geometric intersection numbers.
result Subgroups can be free groups, direct products, or have specific ranks.
Generates spin structure stabilizers using Dehn twists.
problem Determining generating sets for spin structure stabilizers.
method Using Dehn twists on specific curves.
result Explicit generating sets for stabilizers of spin structures.
Boundary Dehn twists become trivial after abelianization.
problem Understanding the behavior of boundary Dehn twists in smooth mapping classes.
method Constructing diffeomorphisms and showing commutators represent boundary Dehn twists.
result Boundary Dehn twists become trivial after abelianization.
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
For odd genus, Dehn twists generate a subgroup with three finite order elements.
problem Understanding the generating set of Dehn twist subgroups in non-orientable surfaces.
method Proving the existence of a generating set of three finite order elements for odd genus non-orientable surfaces.
result For odd genus, the Dehn twist subgroup can be generated by three elements of finite orders.
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
Study connects surface twists to curve invariants.
problem Understanding fractional Dehn twists on Riemann surfaces.
method Characterization of pseudo-periodic maps and bounds on fractional twists.
result Established relationship between twists and curve invariants.
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…
The paper uses Floer homology to study twist coefficients and their behavior after capping off.
problem Behavior of twist coefficients after capping off a boundary component.
method Heegaard Floer homology to constrain twist coefficients.
result Results about fractional Dehn twists and Floer homology of cyclic branched covers.
The paper computes presentations of cluster modular groups and verifies their generation by Dehn twists.
problem Computing presentations and verifying generation of cluster modular groups.
method A method to compute presentations of saturated cluster modular groups and verification of generation by cluster Dehn twists.
result The cluster modular groups of specified types are virtually generated by cluster Dehn twists.
Johnson kernel generated by specific Dehn twists on surfaces.
problem Identifying the generating set of Johnson kernel for surfaces.
method Analyzing Dehn twists on specific types of surfaces.
result Johnson kernel generated by Dehn twists on specified surfaces.
We classify groups generated by powers of 2 Dehn twists which are 1) free or 2) have no ``unexpected'' reducible elements. We give some sufficient conditions in the case of groups generated by powers of more than two twists.
Paper studies involutions generating the twist subgroup of nonorientable surfaces.
problem Generating the twist subgroup by involutions on nonorientable surfaces.
method Analyzes involutions to find the smallest generating sets.
result Provides generating sets of involutions with minimal elements.
The paper explores methods to decompose periodic maps into Dehn twists.
problem Factoring periodic maps into Dehn twists.
method Various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy.
result Existence of conjugates of periodic maps whose product is pseudo-Anosov.
By using a notion of a geometric Dehn twist in ♯k(S2×S1), we prove that when projections of two Z-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the Z-splittings generate a free group of ra…
We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…
Generators found for automorphisms of special groups.
problem Characterizing automorphisms of special groups.
method Using a novel hierarchical shortening argument, the paper provides generators for automorphisms of special groups.
result Generators for automorphisms of special groups are found, including Dehn twists and pseudo-twists.
Study detects if a circuit bounds a disc using curve intersections.
problem Determining if a circuit bounds an embedded disc.
method Analyzing the group generated by Dehn twists about curves in a circuit.
result Cycle relation between Dehn twists detects disc-boundability.
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.
New bounds on categorified link invariants using fractional Dehn twist coefficients.
problem Bounding ranks of categorified link invariants.
method Using fractional Dehn twist coefficients and link Floer homology.
result Lower bounds on ranks of categorified link invariants.
The paper explores generalized Dehn twists on surfaces and their homology cylinders.
problem Understanding and computing generalized Dehn twists on surfaces with self-intersections.
method The paper introduces two ways to generalize Dehn twists and proves their equivalence under certain conditions.
result The automorphism induced by a homology cylinder agrees with the generalized Dehn twist along a curve.
Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.
problem Realization of Dehn twists as finite order diffeomorphisms on spin 4-manifolds.
method Use Y. Kato's 10/8-type inequality for involutions and its refinement.
result Dehn twists about specific spheres in certain 4-manifolds are not homotopic to finite order diffeomorphisms.
Study classifies exceptional surgeries on specific links.
problem Classifying exceptional Dehn surgeries on specific links.
method Examined minimally twisted seven-chain links.
result Classified all exceptional Dehn surgeries on the links.
New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on X1#X2 is not isotopic to identity if determinants r1,r2 are odd. Let Ng denote a closed nonorientable surface of genus g. For g≥2 the mapping class group M(Ng) is generated by Dehn twists and one crosscap slide (Y-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists have nontrivial roots. We gi…
Solves infinite family of cubic polynomial problems.
problem Infinite family of twisted polynomial problems.
method Using Dehn twists and 9-adic expansions.
result Result of twisting depends on 9-adic expansion.
The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.
problem Properties of Dehn twists in smooth 4-manifolds.
method Generalization of a previous result and alternative proof of a consequence.
result Dehn twists along the boundary of simply-connected 4-manifolds are trivial after connected sums.
Study shows Dehn twist coefficients are consistent across different actions on surfaces.
problem Consistency of fractional Dehn twist coefficients under various actions.
method Analyzes left orderings of mapping class groups and uses cofinality properties.
result Fractional Dehn twist coefficients are independent of the underlying action for surfaces with genus > 1.
We construct nontrivial roots of Dehn twists about nonseparating curves.
Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.
problem Understanding the monodromy of Milnor fibrations of surface singularities.
method Using Seiberg-Witten invariant and monopole Floer homology.
result Infinite order non-triviality results for boundary Dehn twists.
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
The study examines Heegaard splittings defined by Dehn twists and finds hyperbolic metrics with specific geodesic lengths.
problem Characterizing Heegaard splittings defined by Dehn twists and their geometric properties.
method Examining Heegaard splittings of genus g≥3 defined by the n-th power of a Dehn twist along a pared acylindrical curve. result For n≥14, the Heegaard splitting has a hyperbolic metric with a closed geodesic of length between 0.7/(n2g2) and 34.3/n2. Study on roots of Dehn twists on nonorientable surfaces.
problem Investigate roots of Dehn twists on nonorientable surfaces.
method Explore existence, conjugacy classes, and maximal degree roots of Dehn twists.
result Prove existence of roots of degree n for sufficiently large g.
Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact s…
Presentations for involutions on non-orientable surfaces up to genus 5.
problem Representing involutions on non-orientable surfaces.
method Dehn twist--crosscap slide presentations.
result Presentations for involutions on non-orientable surfaces of genera up to 5.
Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.
problem Characterizing exotic diffeomorphisms on Milnor fibers.
method Compared family relative Bauer--Furuta invariants of mapping tori.
result Boundary Dehn twist is an exotic diffeomorphism under given conditions.
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…
Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.
problem Determine if the kernel of SO(3) WRT representations is generated by p-th powers of Dehn twists.
method Investigate kernels for different genus and prime p values, showing containment in specific subgroups.
result Kernels are contained in subgroups generated by p-th powers of Dehn twists and other specific elements for certain conditions.
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.
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
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…
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
Boundary Dehn twist on K3 surfaces becomes trivial after abelianization.
problem Understanding the boundary Dehn twist on K3 surfaces. method Obstruction from Baraglia-Konno and global Torelli theorem of K3 surfaces. result Boundary Dehn twist becomes trivial after abelianization.
The positive Dehn twist expressions for the generalizations of the new involutions described in math.GT/0404310 are presented. The homeomorphism types of the Lefschetz fibrations that they define are determined for several examples.