Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
After a short summary of known results on surface-complexity of closed 3-manifolds, we will classify all closed orientable 3-manifolds with surface-complexity one.
We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Ki…
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
Classifies T2-bundles over closed surfaces up to isomorphisms.
problem Classifying T2-bundles over closed orientable surfaces. method Using group homomorphisms and mapping class groups.
result Any orientable T2-bundle over Σg with g≥1 is isomorphic to the fiber connected sum of g pieces of T2-bundles over T2. Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.
problem Embedding hyperbolic surfaces into hyperbolic 3-manifolds with specific symmetries.
method Examined orientation-preserving and orientation-reversing actions on surfaces, including nonorientable ones.
result Found conditions for equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds.
Algorithm calculates genus of embedded graphs on surfaces.
problem Finding minimal and maximal genus for graph embeddings.
method Constructing special branched coverings of the 2-sphere.
result Algorithm calculates orientable genus and maximal genus of graphs.
Classifies 3-manifolds from cube identifications.
problem Classifying non-orientable 3-manifolds.
method Identifying cube faces to form manifolds, classifying based on surface-complexity.
result Identifies four flat non-orientable 3-manifolds with surface-complexity one.
We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…
We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.
We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M, F is isotopic to a least-area surface F(g), then F is incompressible.
We introduce colorings of oriented surface-links by biquasiles using marked graph diagrams. We use these colorings to define counting invariants and Boltzmann enhancements of the biquasile counting invariants for oriented surface-links. We provide examples to show that the invariants can distinguish both closed surface…
For a closed surface S, its Torelli group I(S) is the subgroup of the mapping class group of S consisting of elements acting trivially on H1(S;Z). When S is orientable, a generating set for I(S) is known. In this paper, we give a normal generating set of I(Ng) for …
Researchers classify and visualize 5-cube cubical surfaces.
problem Classifying and visualizing surfaces in a 5-dimensional cube.
method Exhaustive search, classification by genus and demigenus, 3D visualization, reinforcement learning for optimization.
result 2690 connected closed cubical surfaces in the 5-cube, visualized and optimized for 3D printing.
The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.
problem Dynamics of group actions on SU(2)-representation varieties of surfaces and non-orientable surfaces.
method Description and analysis of group actions generated by Dehn twists on SU(2)-representation varieties.
result Explicit invariant rational functions on SU(2)-representation varieties for specific cases of surfaces and non-orientable surfaces.
Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-ori…
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.
Penner coordinates are extended to the Teichmüller spaces of oriented closed surfaces.
Study non-orientable surfaces to find loops winding around punctures.
problem Understanding loops winding around punctures on non-orientable surfaces.
method Analyzing one-sided geodesics and their powers on non-orientable surfaces.
result Loops can wind around a puncture at most two values of m for odd m. 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. The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.
problem Existence of surfaces of section for geodesic flows on closed surfaces.
method Study of configurations of simple closed geodesics and use of the curve shortening flow.
result Construction of surfaces of section that intersect or have hyperbolic components in their boundary.
Characterizes when curves form bouquets in surfaces.
problem Understanding when simple closed curves form bouquets in surfaces.
method Characterization in terms of relations between Dehn twists.
result Characterization of bouquets of curves in surfaces.
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…
We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi-Kuno and Masseyeau-Turaev.
Study on minimal surfaces with closed curvature lines in 3D space.
problem Investigating complete non-orientable minimal surfaces with specific curvature properties.
method Analyzing complete non-orientable minimal surfaces of finite total curvature in R3 with ends foliated by closed lines of curvature. result There are no such surfaces with one end, proving a rigid situation.
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
Study shows mapping class group dimension for surfaces with punctures.
problem Determining the geometric dimension of mapping class groups of surfaces with punctures.
method Proved cocompact classifying space for proper actions with dimension equal to virtual cohomological dimension.
result Proper geometric dimension of mapping class groups of orientable surfaces with punctures.
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves. This is a variant on an earlier construction of Bestvina-Bux-Margalit.
The systole size affects the smallest eigenvalue of hyperbolic surfaces.
problem Determining the smallest eigenvalue of hyperbolic surfaces based on their systole size.
method Analyzing the relationship between systole size and the smallest eigenvalue of Laplacian on hyperbolic surfaces.
result If the systole of a hyperbolic surface is greater than 3.46, then the smallest negative eigenvalue is greater than 1/4.
For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.
We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear repres…
Partial coverings of hyperbolic surfaces equidistribute with geodesics.
problem Equidistribution of partial coverings defined from geodesics.
method Sequence of geodesics equidistributing in unit tangent bundle implies equidistribution of associated partial coverings.
result Partial coverings equidistribute with a sequence of geodesics.
Estimates the number of closed curves on surfaces with power-saving error terms.
problem Counting closed curves on surfaces with given properties.
method Effective dynamics of mapping class group on Teichmüller space and space of closed curves, introducing novel methods.
result Proves estimates with power-saving error terms for filling closed curves and curves with respect to a current.
Finite groups act freely on surfaces but not on 3-manifolds.
problem Understanding finite group actions on surfaces and 3-manifolds.
method Analyzing homeomorphisms of surfaces and 3-manifolds.
result Finite groups can act freely on surfaces but not on handlebodies.
Analog of Kauffman bracket for non-orientable knots in thickened surface.
problem Defining an invariant for non-orientable knots in a non-orientable surface.
method Proposes an analog of the Kauffman bracket polynomial with modified sign rules.
result Polynomial is an isotopy invariant and independent of classical Kauffman for orientable covers.
Unified framework for 3D and 4D manifold and knot theory.
problem Unified understanding of manifold and knot theory.
method Unified correspondence between different subfields of low-dimensional topology.
result Unified algebraic manifestations of 3D and 4D manifold and knot theory.
The paper solves conditions for extending circle-valued Morse functions.
problem Conditions for extending circle-valued Morse functions on closed orientable surfaces.
method Provided necessary and sufficient conditions for the existence of a non-singular extension.
result Necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function.
We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all π1 of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollar…
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.
New method uses non-orientable surfaces to describe knots in 3-manifolds.
problem Describing and understanding knots in 3-manifolds.
method Defining a plat-like representation based on embedded non-orientable surfaces.
result Any link in the manifold can be represented as a plat-like closure of a surface braid group element.
Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
Holomorphic handle attaching proves complex surface properties.
problem Characterizing complex surfaces with contact boundaries.
method Holomorphic handle attaching method.
result Closed contact 3-manifolds can be filled as complex surfaces' boundaries.
Paper studies R∞-property for braid groups on surfaces, proving it for most cases.
problem Investigating the R∞-property for braid groups on orientable surfaces.
method Analyzing surface pure and full braid groups to determine the R∞-property.
result Most surface braid groups have the R∞-property, with exceptions noted.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.