Paper calculates L-invariant and L*-invariant for complex surface sums.
problem Calculating invariants for complex surface sums.
method Using pants complexes and dual curve complexes.
result First example of arbitrary large invariants for bridge numbers.
Formula connects surface and curve invariants via slice transitions.
problem Computing surface invariants from curve invariants.
method Introducing differential measures for local changes across singular slice transitions.
result Explicit formula for surface invariant change during quadruple-point events.
A surface in homogenous space Sol is said to be an invariant surface if it is invariant under some of the two 1-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classi…
In this paper, we formulate a construction of ideal coset invariants for surface-links in 4-space using invariants for knots and links in 3-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of sur…
Yamabe invariants of certain non-Kähler surfaces are zero.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
New method to determine parabolic surfaces invariant under Killing fields.
problem Characterizing parabolic invariant surfaces.
method Clear and practical characterization method for invariant surfaces.
result A new way to determine parabolicity of invariant surfaces.
Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.
problem Identifying compact surfaces up to rigid transformations.
method Degree four polynomials in moments of delta function, effective inversion algorithm.
result Invariants and retrieval algorithm work on a comeagre subset of surfaces.
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Turaev-Viro invariants match for certain surface bundles.
problem Matching Turaev-Viro invariants for surface bundles.
method Profinite completions and spherical fusion categories.
result Invariants coincide for specific surface groups.
We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…
Study structural invariants of Goursat distributions related to curve singularities.
problem Understanding local invariants of Goursat distributions.
method Investigate structural invariants akin to curve singularities on surfaces.
result Relate structural invariants to small growth invariants in the sequel.
New minimal surfaces found in a specific type of 3D space.
problem Finding minimal surfaces in a particular class of 3D spaces.
method Investigated minimal surfaces invariant under a specific group action in unimodular semidirect products.
result Described new examples of minimal surfaces in a specific 3D space.
Upper bounds for surface-links in the Yoshikawa table are estimated.
problem Estimating Kirby-Thompson invariants of surface-links.
method Using tri-plane diagrams and L-, L*-invariants.
result Upper bounds for surface-links in the Yoshikawa table are obtained.
The paper studies invariants of surfaces in the 3-sphere using handlebody-links.
problem Understanding invariants of surfaces in the 3-sphere.
method Using Heegaard splittings and G-families of quandles to construct invariants. result Invariants can distinguish certain surfaces in the 3-sphere.
Extends Milnor invariants to surface-links using cut-diagrams.
problem Tackles invariants for surface-links in 4-space.
method Introduces cut-diagrams and groups associated to them.
result Yields concordance and link-homotopy invariants for surface-links.
New invariant for Riemann surfaces connects to moduli space classes.
problem Understanding Riemann surface invariants and their relationships.
method Introducing a twisted version of the Kawazumi-Zhang invariant and relating it to other classes.
result The new invariant connects to the first Mumford-Morita-Milller class on the moduli space.
In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in R3. In particular, we show how to parametrize a loxodrome on an invariant surface of $…
The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any …
Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.
problem Unbounded stabilization heights of fiber surfaces.
method Alternative proof using Hopf invariant.
result Proves unbounded stabilization heights of fiber surfaces.
Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual lin…
We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariant…
New invariant for hyperbolic surfaces, geometric criterion for domains.
problem Geometric criterion for bounded domains in complex plane.
method Renormalized volume type invariant on hyperbolic surfaces.
result New geometric criterion for bounded domains in complex plane.
Study invariants of null-homologous knots in thickened surfaces.
problem Understanding concordance properties of knot invariants.
method Using Gordon-Litherland pairing and cobordism results for closed unoriented surfaces.
result Brown invariants are invariant under concordance of spanning surfaces.
Sharp bounds for Kirby-Thompson invariants of knotted surfaces computed.
problem Computing sharp lower bounds for Kirby-Thompson invariants of knotted surfaces.
method Using dual curve complex distances to compute invariants.
result Exact values of KT-invariants computed for knotted surfaces with bridge number ≤ 6.
Invariants defined for braid systems under Hurwitz equivalence.
problem Invariants for braid systems under Hurwitz equivalence.
method Crossing matrix and polynomial invariant introduced.
result Invariant defined for surface braids and surface links.
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…
Enhanced bikei modules distinguish unoriented and non-orientable surface-links.
problem Distinguishing unoriented and non-orientable surface-links.
method Extending biquandle module invariants to unoriented surface-links using bikei modules.
result Enhanced bikei modules are more effective at distinguishing non-orientable surface-links than bikei homset cardinality alone.
In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the sh…
Introduces fundamental heaps for surfaces, linking them to cocycle invariants.
problem Isotopy invariants of surface embeddings in 3-space.
method Definition of fundamental heaps using surface ribbons and heap operations.
result Fundamental heaps have a free part whose rank matches the number of connected components.
The paper proves a generalized Bour's theorem for invariant surfaces.
problem Proving a generalized Bour's theorem for invariant surfaces.
method Equivariant geometry techniques applied to three-manifolds.
result A generalized Bour's theorem holds for invariant surfaces.
New surface observables yield 2-knot invariants in nonabelian theories.
problem Developing new invariants for nonabelian theories.
method Introducing surface observables in BF theory and Yang-Mills theory.
result Surface observables induce new 2-knot invariants and electric fluxes.
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
New invariants for virtual knots via spanning surfaces.
problem Tackling invariants for virtual knots and distinguishing them from classical knots.
method Defining new invariants using spanning surfaces and Seifert matrices.
result An Alexander polynomial for virtual knots that can distinguish knots from their mirrors.
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
Explains Milnor invariants for links and surfaces.
problem Milnor invariants of links and surfaces.
method Expository viewpoint of research.
result Translation and explanation of Milnor invariants.
Extends Gordon-Litherland pairing to links in thickened surfaces, defining new invariants.
problem Defining invariants for links in thickened surfaces.
method Extending Gordon-Litherland pairing, defining new invariants based on spanning surfaces.
result Invariants depend only on S∗-equivalence class of spanning surfaces and give well-defined invariants of virtual links. The paper extends a formula linking surface curvature to projection invariants.
problem Investigating invariants of projected surface curves.
method Introduced invariants of plane curves from surface projections; extended d'Ocagne formula.
result Extended d'Ocagne formula connects surface curvature with projection behavior.
The paper studies invariant functions and their relation to Landsberg surfaces.
problem Investigating the geometry of invariant functions and their applications to Landsberg surfaces.
method Investigating the geometry of S-invariant functions and their associated vertical subdistribution, and relating the holonomy distribution to these subdistributions. result For Landsberg surfaces, if the flag curvature is S-invariant, it is constant, and the surface is Riemannian. The study defines invariants for time-like surfaces with real asymptotic lines.
problem Characterizing time-like surfaces with real asymptotic lines.
method Fundamental theorem of Bonnet-type, canonical parameters, invariant functions, PDEs.
result Time-like surfaces are determined by four invariant functions, two of which can be Gauss and mean curvature.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
Study of small growth invariants in Goursat distributions.
problem Understanding local invariants of Goursat distributions.
method Analysis of the small growth sequence and its relation to structural invariants.
result Relate small growth invariants to structural invariants of Goursat distributions.
Classifies solitons on invariant surfaces in solvable Lie group.
problem Classifying solitons on invariant surfaces in a specific Lie group.
method Analyzes solitons associated with Killing vector fields and proves rigidity results.
result Identifies the only solitons for specific invariant surfaces.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
We study the second order invariants of a Lorentzian surface in R2,2, and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrin…
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space E41. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface…
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.